Dynamical behaviour and meta heuristic optimization of a hospital management software system

Abstract

The growing demand for easily available healthcare in recent years has fuelled the digitization of healthcare services. The Hospital Management System (HMS) software stands out as a comprehensive solution among the software systems and tools that hospitals and clinics are developing in tandem with this trend. In order to effectively manage many facets of hospital operations, in this paper, we propose an approach for investigating software of this kind. Thus, we characterise the HMS software as a unique sort of batch arrival retrial queueing system (QS) that can handle both ordinary and priority patient demands. Furthermore, it permits patient resistance (balk) and departure (renege) in specific circumstances. The proposed model is additionally deployed within the framework of Bernoulli working vacation. The supplementary variable technique (SVT) has been utilised to obtain the necessary results. ANFIS, a soft computing tool, is used to validate the analytical results as well. Finally, this study seeks to enhance the cost-effectiveness of software creation by employing four unique optimization methods, aiming to achieve optimal efficiency in resource utilization.

1. Introduction

In today's environment, patients' smartphones serve as the first point of contact in healthcare management, allowing them to express their own preferences and request services whenever and wherever they are needed. Moreover, there has been a dramatic rise in the software industry's need to supply highly secure software. It is also pointless to provide user-friendly software without proper protections. As a result, it's critical to offer accessible software services for defence in order to find optimal solutions and close the gap between the two. To bridge the gap between usability and safety, numerous studies on useable security evaluations have been conducted. The HMS software provides a streamlined method for operating the various hospital facilities as a whole and addressing its many operational challenges. Thus, the emergence of the HMS software was in response to the pressing necessity to facilitate the acceleration of hospital procedures. A HMS softwares, is a web-based medical management infrastructure that operates autonomously or in the cloud and collects and combines data from all departments. HMS software are designed to improve care delivery by streamlining clinician workflow, decreasing overhead and mistake rates, and enhancing the quality of life for patients. This system oversees every department or branch of the hospital, therefore it involves a vast array of tasks. A brief depiction of HMS is given in Fig. 1. Data such as lab results, medical supplies, appointments, bills, insurance, and reports are compiled by it.

Figure 1.

Hospital management system.

1.1. Insights into hospital management systems: recent perspectives

In recent years, numerous researchers have extensively investigated HMS, especially with the burgeoning integration of Internet of Things (IoT) technology in healthcare administration. This integration has brought forth groundbreaking solutions to various healthcare challenges. For example, Rizk et al. [28] developed a SMART HMS aimed at interpreting health data to enhance patient treatment quality. Their focus was on delineating the essential requirements for IoT-enabled healthcare systems. Their approach delineated a set of functional and non-functional prerequisites alongside their implementation strategies to achieve optimal healthcare system performance. Additionally, Uslu et al. [34] conducted a thorough examination of factors influencing smart hospital design leveraging IoT technology. They acted as a bridge between computerized nodes in heterogeneous networks and business applications. Moreover, in light of the convergence of Artificial Intelligence (AI) and IoT, Aal et al. [1] introduced a novel paradigm aimed at elevating the standard of care in smart healthcare systems. To further refine the IoT-based SHS model, they introduced the particle swarm optimization-long short-term memory (PSO-LSTM) technique.

Shala et al. [29] explored various digital solutions to enhance hospital administration by creatively integrating blockchain technology to safeguard and streamline Electronic Medical Records (EMRs) management. They emphasize the importance of continuous monitoring, quality assessment, and the usability and efficiency of these innovations. Ala et al. [2] analyzed a sustainable healthcare supply chain network and developed a resilient fuzzy technique to address parameter uncertainty, alongside proposing a multiple objective differential evolution algorithm (MODEA) to solve the fuzzy model. Additionally, Ala et al. [3] investigated the use of Mixed Integer Linear Programming (MILP) for patient appointment scheduling in hospitals, and again they [4] applied MILP and an approximate solution approach (ASA) to address the health chain problem, utilizing QS framework. Moreover, Ala et al. [5] conducted a thorough review of healthcare simulation appointment scheduling systems, identifying key challenges effectively modelled through simulation techniques. Thus, extensive research has been conducted on HMS.

1.2. Queueing variability in retrial queues frameworks

Conversely, waiting for a service is a common occurrence in daily life, serving various important purposes within processes. The concept of retrial queues (RQ) arises from this natural phenomenon. In a retrial QS, when a customer is denied service, they enter a waiting orbit and attempt again from there. This significantly adds complexity to the system, making the analysis of RQ challenging. Such queues have wide-ranging implications across different fields, including call centers, retransmission-enabled communication networks, and more.

Furthermore, research on two categories of clients namely ordinary and priority has been carried out extensively in the past. High priority consumers are either served in a queue or not, depending on the pre-emptive or non-pre-emptive discipline. Clients with low priority, or ordinary clients, who have been obstructed depart the system and enrol in the retrial group to retry their service when the server is accessible. A few works on priority queues are follows, Xu et al. [37] assessed a $M/G/1$ RQ with priority service that was subjected to multiple vacation periods. Implementing the SVT, Nila and Sumitha [21] dealt with a consignment advent RQ version with pre-emptive priority. Lan and Tang [18] probed the RQ under probabilistic pre-emptive priority through discrete time $Geo/G/1$. Furthermore, in QS, impatience among clients is also a common phenomenon. Balking occurs when clients choose not to join the queue upon arrival if the server is unavailable, while reneging happens when clients leave the queue after waiting too long for service. Recent research has explored various aspects of impatient clients as well. For instance, Nila and Sumitha [22] investigated a RQ with bulk arrivals, dual essential services and inconsistent consumers. Similarly, Rajasudha et al. [26] analyzed a discrete-time RQ with bulk arrivals, impatient consumers under three types of vacation policies.

In QS, vacation periods refer to temporary breaks in server activity, while working vacation (WV) periods involve servers providing service at a reduced rate compared to regular operations. GnanaSekar and Kandaiyan [9] explored a single server RQ with inconsistent clients deployed under WV. Recently, Rajadurai [24] scrutinized a sole server pre-emptive priority RQ with Bernoulli working vacation (BWV) and vacation interruption. Recently, an investigation on the sole server priority retrial G-queue with BWV was carried out by Rajadurai [25]. Divya et al. [7] employed stochastic variability techniques to analyze a RQ with pre-emptive priority arrivals during WV, providing optimal solutions for system performance metrics. Upadhyaya and Shekhar [33] evaluated a priority RQ queue with batch arrivals under working breakdown and WV scenarios using the maximum entropy approach.

1.3. Exploring retrial queues dynamics in healthcare management

In the aforementioned literature review provided, we have exclusively focused on research pertaining to the RQ framework. However, several studies have delved into RQ models within the context of healthcare systems. Zhu and Liu [39] recently conducted an analysis of a feedback patient flow system, treating it as a RQ structure within a Chinese outpatient facility. Gao and Zhang [17] utilized SVT to examine an $M/G/1$ RQ operating under server vacation scenarios, proposing a modeling approach for hospital service systems with potential applications in healthcare problem-solving. Bhagat and Jain [8] explored a bulk arrival non-Markovian system featuring multiple repairs and services, using SVT to analyze the retrial framework with a focus on non-pre-emptive priority contexts relevant to healthcare systems. Rani et al. [27] investigated a Markovian RQ with double orbits, orbital search, and disaster scenarios. They employed the golden section search method to optimize service parameters for maximizing profit, suggesting that their framework could be adapted as a model for healthcare systems, offering solutions to real-world challenges.

1.4. Insight into cost optimization

Kolahi et al. developed a simulation optimization system, as detailed in their study [15], aimed at evaluating and improving performance factors within multi-level and multi-commodity supply chains. Ala et al. [6] approached the dynamic capacitate facility location problem in mobile charging stations from a sustainability perspective, presenting a two-stage stochastic programming approach alongside scenario-based optimization techniques to validate their efficacy [6]. However, there remains a scarcity of research concerning cost optimization in RQ and the assessment of appropriate control parameters for queuing models using various optimization techniques. Anshul Kumar and Madhu Jain examined a Markovian QS [16], formulating a cost function to determine the optimal service rates that minimize costs through a combination of two distinct optimization methodologies. Additionally, for an overview of works on cost optimization, readers may refer to articles such as ([35], [36], and [12]). While research on RQ within healthcare systems exists, from the aforementioned literature review, there are significantly very few investigations that concentrate on batch arrival RQ that are exposed to BWV, especially when it comes to describing HMS software. Considering this rationale, the study's primary contributions are as follows:

• •By presenting an HMS program modelled as a batch arrival pre-emptive priority RQ with impatient clients deployed under BWV, the research fills a major void in the literature thereby presenting a unique approach.
• •To ensure the robustness and validity of the findings, the study employs Adaptive Neuro-Fuzzy Inference System (ANFIS) for result validation and assurance regarding the accuracy of the suggested model.
• •Finally, the utilization of four different optimization techniques for cost-effectiveness analysis in order to enhance the practical applicability of the proposed HMS, offering insights into resource allocation and management strategies for healthcare providers.

1.5. Implementation of the suggested framework into HMS software

Here, the framework under consideration has an intriguing application in HMS software which enables the hospitals with the means to manage records and data pertaining to all facets of healthcare. Thus, the proposed model has been modelled as follows, The HMS software acts as a regular server since it is highly essential due to the large volume of data, the number of people involved, and the sheer number of processes. The benefits of implementing an HMS encompass improved patient care, appointment management, insurance and payment data storage, access to patient records for both doctors and clinicians, and the ability to ensure care consistently (services). The aforementioned services can be accessed at any time by patients (ordinary patients) via the software, while the respective hospitals' staff (priority patients) can preside over the services a little bit sooner and with some benefits. When multiple requests are submitted to a server at once, the server will only handle one at a time if it is not preoccupied. Then, the existing requests will be redirected back to a buffer area, which is a kind of virtual waiting area (orbit). In addition, there's a possibility that patients may opt out (balk) of the service or will bail out in the middle (renege) of it if the server or network connection falters. In the absence of a patient requests, the HMS software will execute a variety of routine tasks, such as a virus scan (multiple WV). Moreover, in the mainstream data centre, various subsystems, such as the computerised order entry (COE) and the management scheduling distributor (MSD) (WV server), handle requests at a reduced rate (WV) while the primary data centre is down for maintenance (WV period). Hence, HMS tends to produce a healthcare model that is distinct, efficacious, rapid, and considerate, and that would resonate with a wide range of people.

1.6. Research novelty

Numerous studies have explored healthcare systems using various methodologies, including fuzzy logic ([2]) optimization techniques ([3], [4]), and IoT technology ([28], [34], [1]). Additionally, many researchers have focused on queue management systems in hospitals, with some exploring applications in retrial QS within hospital service systems ([39], [17], [8]). However, there has been a gap in the literature regarding RQ frameworks characterizing HMS software. Further, though there are only some studies have investigated single-server QS under BWV using the SVT ([24], [25]), limited attention has been given to batch arrival RQ under BWV.

Motivated by this, in this study, we address a generalization of the single and multiple WV queueing models (i.e.) BWV with the concept of RQ by characterizing a HMS software into a unique bulk arrival queue. Therefore, by incorporating the concepts of bulk arrival and impatient consumers, we have broadened the work of Rajadurai [24]. As far as the author is aware, hardly any research has been addressed in the queueing domain that characterizes an HMS software as a batch arrival RQ with pre-emptive priority and impatient consumers deployed under BWV by utilizing the concept of SVT.

1.7. Research goal

The primary objective of this work is to use the SVT to calculate the probability-generating functions and steady-state probabilities for different HMS service states. Next, some numerical examples should be used to illustrate how varying some parameters affects the system as a whole. Moreover, ANFIS will be utilized for validating the analytical outcomes by comparing them to those of the neuro-fuzzy method. Furthermore, this system employs a variety of optimization techniques in an effort to achieve the optimal possible total cost.

The subsequent content describes how the residual objectives are ordered: In Section 2, a thorough reasoning of the framework under consideration. Section 3 covers the steady state (SS) findings, alongside the system stability criterion. Section 4 covers a number of noteworthy system measurements. The ANFIS results have been contrasted with the statistical analysis in Section 5, alongside the influence of distinct attributes on the system's framework. In Section 6, four separate optimization strategies are employed to analyze cost optimization. A comprehensive analysis of the suggested work is presented in Section 7.

2. The model's overview

In this article, a stochastic model of a HMS software has been modelled as a batch arrival pre-emptive priority RQ with impatient consumers under BWV. An illustration of the model under consideration and its pictorial depiction (Fig. 2) have been presented below. In addition, the transition diagram of the model is also presented in Fig. 3.

Figure 2.

A pictorial depiction of the model.

Figure 3.

The transition diagram of the model.

Arrival process: Patients requests for services in bulk in two different categories: priority patients and regular patients. In the service time of a busy server, priority patients are given precedence over regular ones. Assume that the bulk arrival of both priority and regular patients follows two separate ‘‘Poisson processes’’with paces ζ and ϖ respectively. Let ${\mathcal{W}}_k$ indicate the no. of patients related to the $k^{th}$ arrival group, in which $k=1,2,3,...$ are distributed arbitrarily. ‘‘$Pr[{\mathcal{W}}_k=n]={\vartheta }_n,n=1,2,3,...$’’and $\mathcal{W}(\breve{\xi })$ signifies the probability generating function (PGF) of $\mathcal{W}$.

Retrial process: If an approaching priority patient realises the server is vacant, then their service starts right away. The arrival of a priority request will pause the service of an ordinary request by the regularly busy server, and the server will then start providing service to the priority patient. In the case that a priority patient took precedence over an ordinary patient. We predicted that ordinary patient, had just been served prior to the priority patient's service commencement, would remain in the service station until the remaining service was accomplished.

In the instance that a patient request for a service and discovers that the server is unoccupied, they are free to immediately begin using it as long as there is no designated waiting space. If they request and find the server busy or on WV, they are required to either exit the system with prob. $(1-b)$ or join the buffer area with a group of obstructed requests with prob. b in compliance with first come first serve $(FCFS)$ discipline. As soon as the server stops processing requests, the patient at the top of the RQ engages in a battle of wits with other possible primary patients to decide who will be given access to the service next. If a regular or priority patient enters before the retrial client, the latter may cancel its request for service and either return to its previous position in the RQ with prob. s or exit the system with prob. $(1-s)$. In this case, an arbitrary dist. $\mathcal{D}(\stackrel{`}{\upsilon })$ with a subsequent Laplace Stieltjes Transform (LST) ${\mathcal{D}}^{⁎}(t)$ is used to standardise the retrial duration for all orbiting patients.

Regular Service process: The server immediately begins normal service whenever a new primary (priority) patient or retrial patients arrives at the server in an idle state. Priority patients' service times are distributed generally and are represented by the random variable ${\mathcal{L}}_p$ with the dist. fn. ${\mathcal{L}}_p(\stackrel{`}{\upsilon })$ possessing LST ${\mathcal{L}}_p^{⁎}(t)$ and the first and the second moments are given by ${\mu }_p^{(1)}$ and ${\mu }_p^{(2)}$. Whereas the service times of the regular patients are also distributed generally and are represented by the random variable ${\mathcal{L}}_b$ with the distribution function ${\mathcal{L}}_b(\stackrel{`}{\upsilon })$ having LST ${\mathcal{L}}_b^{⁎}(t)$ and the first and the second moments are given by ${\mu }_b^{(1)}$ and ${\mu }_b^{(2)}$.

Bernoulli Working Vacation process: Every time the buffer (orbit) goes vacant, the server starts a WV, which has an exponential distribution with a parameter γ. If any new patients come in on vacation, the server keeps running but at a slower speed. The WV period is a slower-moving operational period. If any orbiting patients with significantly slower completion periods complete their service during the vacation period, the server will terminate the vacation and resume the usual busy period. This results in a vacation interruption. Otherwise, if there are no clients in the system at the end of the vacation, the server either enters the system and waits passively for a new patient request with prob. p (single WV) to serve the patients as they arrive in regular mode, or it departs for another WV with prob. $q=1-p$ (WV). If there are still patients in the buffer area when a vacation expires, the server resumes regular functioning. During the WV period, the service time is represented by the random variable ${\mathcal{L}}_v$ with the distribution function ${\mathcal{L}}_v(\stackrel{`}{\upsilon })$ having LST ${\mathcal{L}}_v^{⁎}(t)$ and the first moment is given by ${\mathcal{L}}_v^{{⁎}^{\prime }}(\gamma )={\int }_0^{\infty }\stackrel{`}{\upsilon }e{-}^{\gamma \stackrel{`}{\upsilon }}d{\mathcal{L}}_v(\stackrel{`}{\upsilon })$.

The system's stochastic processes are all believed to be independent of one another.

Further, the tail of the dist. fn. $\mathcal{F}(\stackrel{`}{\upsilon })$ is referred all through the rest of the article as $\bar{\mathcal{F}}(\stackrel{`}{\upsilon })=1-\mathcal{F}(\stackrel{`}{\upsilon })$. In addition, the LST of $\mathcal{F}(\stackrel{`}{\upsilon })$ and the Laplace transform of $\mathcal{F}(\stackrel{`}{\upsilon })$ are denoted by

$${\mathcal{F}}^{⁎}(n)=\underset{0}{\overset{\infty }{\int }}{e}^{-}n\stackrel{`}{\upsilon }d\mathcal{F}(\stackrel{`}{\upsilon })\text{and}{\tilde{\mathcal{F}}}^{⁎}(n)=\underset{0}{\overset{\infty }{\int }}{e}^{-}n\stackrel{`}{\upsilon }\mathcal{F}(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }\text{respectively}$$

Also, we assume the notion,

$${\bar{\mathcal{F}}}^{⁎}(n)=\frac{1-{\mathcal{F}}^{⁎}(n)}{n}.$$

3. System's steady state analysis

A detailed analysis of the proposed model is presented in the previous section. Next, we define corresponding hazard rate functions for the various system states. Then, to clearly construct the proposed framework, we define its different state of the server depending on the patients availability. Before framing the SS diff. eqns. for the various system states, we prove the ergodicity of the model as well. Also, by treating the elapsed retrial times, elapsed service times of the priority/ordinary patients, elapsed WV times as supplementary variables, we formulate the SS diff. eqns. and its corresponding boundary conditions with the aid of SVT. Finally, we make use of the generating function method to derive the SS solution of the suggested RQ framework thereby finding the PGF of the no. of patients in the system as well the orbit.

3.1. Probabilities and notations for the steady state

In SS, we consider that $\mathcal{D}(0)=0$, $\mathcal{D}(\infty )=1$, ${\mathcal{L}}_p(0)=0$, ${\mathcal{L}}_p(\infty )=1$, ${\mathcal{L}}_v(0)=0$, ${\mathcal{L}}_v(\infty )=1$, are continuous at $\stackrel{`}{\upsilon }=0$ and ${\mathcal{L}}_b(0)=0$, ${\mathcal{L}}_b(\infty )=1$, are continuous at $\tilde{\varsigma }=0$. Therefore, the functions $\phi (\stackrel{`}{\upsilon })$, ${\chi }_p(\stackrel{`}{\upsilon })$, ${\chi }_b(\tilde{\varsigma })$ and ${\chi }_b(\stackrel{`}{\upsilon })$ are the ‘‘conditional completion rates (hazard rates)’’for retrial, service of a priority and ordinary patients, lower rate service respectively.

$$\phi (\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }=\frac{d\mathcal{D}(\stackrel{`}{\upsilon })}{1-\mathcal{D}(\stackrel{`}{\upsilon })};{\chi }_p(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }=\frac{d{\mathcal{L}}_p(\stackrel{`}{\upsilon })}{1-{\mathcal{L}}_p(\stackrel{`}{\upsilon })};{\chi }_b(\tilde{\varsigma })d\tilde{\varsigma }=\frac{d{\mathcal{L}}_b(\tilde{\varsigma })}{1-{\mathcal{L}}_b(\tilde{\varsigma })};{\chi }_v(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }=\frac{d{\mathcal{L}}_v(\stackrel{`}{\upsilon })}{1-{\mathcal{L}}_v(\stackrel{`}{\upsilon })}.$$

In addition, let ${\mathcal{D}}^0(\epsilon )$, ${\mathcal{L}}_p^0(\epsilon )$, ${\mathcal{L}}_b^0(\epsilon )$ and ${\mathcal{L}}_v^0(\epsilon )$ be the elapsed retrial times, elapsed service time of the priority patient, elapsed service time of the ordinary patient, elapsed WV times respectively at time ε.

3.2. The proposed model's steady state equations

Let's define the random variable for the construction of this retrial QS,

$$\mathrm{\Phi }(\epsilon )=\{\begin{array}{cc}0,\hfill & \text{the server being inactive and in WV period}\hfill \\ 1,\hfill & \text{the server being available and in normal service mode}\hfill \\ 2,\hfill & \text{the server being occupied with a priority patient without pre-empting }\hfill \\ \hfill & \text{an ordinary patient request and in normal service mode at time}\mathit{\text{ε}}\hfill \\ 3,\hfill & \text{the server being occupied with a priority patient with pre-empting}\hfill \\ \hfill & \text{ an ordinary patient request and in normal service mode at time}\mathit{\text{ε}}\hfill \\ 4,\hfill & \text{the server being occupied with an ordinary patient request and in regular}\hfill \\ \hfill & \text{service period at time}\mathit{\text{ε}}.\hfill \\ 5\hfill & \text{the server being occupied and in WV period}.\hfill \end{array}$$

Additionally, we emphasise how the bivariate Markov process {$\mathrm{\Phi }(\epsilon )$, $\mathrm{\Theta }(\epsilon )$; ε ≥ 0} may be used to classify the system's state at time ε, where $\mathrm{\Phi }(\epsilon )$ signifies the server state $(0,1,2,3,4,5)$ based on whether the server is inactive, engaged for priority patients, regular patients, regular feedback patients and WV. $\mathrm{\Theta }(\epsilon )$ indicates the no. of patients in the orbit at time ε. ${\mathcal{D}}^0(\epsilon )$ correlates the elapsed retrial time, when $\mathrm{\Phi }(\epsilon )=1$ and $\mathrm{\Theta }(\epsilon )>0$. If $\mathrm{\Phi }(\epsilon )=2$ and $\mathrm{\Theta }(\epsilon )\ge 0$, then ${\mathcal{L}}_p^0(\epsilon )$ correlates the elapsed service time spent serving the priority patient in regular busy period. If $\mathrm{\Phi }(\epsilon )=3$ and $\mathrm{\Theta }(\epsilon )\ge 0$, then ${\mathcal{L}}_p^0(\epsilon )$ correlates the elapsed service time spent serving the interrupted regular patient in regular busy period. If $\mathrm{\Phi }(\epsilon )=4$ and $\mathrm{\Theta }(\epsilon )\ge 0$, then ${\mathcal{L}}_b^0(\epsilon )$ correlates the elapsed service time spent serving regular patient in regular busy period. If $\mathrm{\Phi }(\epsilon )=5$ and $\mathrm{\Theta }(\epsilon )\ge 0$, then ${\mathcal{L}}_v^0(\epsilon )$ correlates the elapsed time of the patient spent serving in lower rate service period.

3.2.1. Model's ergodicity condition

At the departure or vacation epochs, we examine the embedded Markov chain's (MC) ergodicity. The notation {${\epsilon }_n;n=1,2,\dots$} denotes the list of epochs during which a regular service for priority/ordinary or a lower service fulfilment occurs. Thus, an embedded MC in the retrial QS is formed by utilizing a sequence of random vectors $A_n=\{\mathrm{\Phi }({\check{\epsilon }}_n+),\mathrm{\Theta }({\check{\epsilon }}_n+)\}$ and its state space is given as $S=(0,1,2,3,4,5)\times \mathrm{\Theta }$.

Theorem 3.1

The embedded Markov chain(MC)$\{A_n;n\in N\}$is ergodic iff$\rho <{\mathcal{D}}^{⁎}(\zeta +\varpi )$, where,

$$\rho =({\mathcal{D}}^{⁎}(\zeta +\varpi )+\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}]+\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))\zeta b{\mathcal{I}}_1{\mu }_p^{(1)}-\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )s{\mathcal{I}}_1$$

Proof

By using Foster's criteria [23], it is straightforward to confirm that ergodicity is a sufficient condition which further suggests that the MC $\{A_n;n\in N\}$ may be irreducible and aperiodic. For a MC to be ergodic, there must be a non-negative function $g(d)$, in which $d\in N$ and $ϵ>0$, such that the mean drift ${\beta }_d=E[g(c_{n+1})-g(c_n)/c_n=d]$ is limited for all $d\in N$ and ${\beta }_d\le -ϵ$ for all $d\in N$, with the possible exception of some finite number of $d^{\prime }$s. If we take into account the fn. $g(d)=d$ in this context, we get:

$${\beta }_u=\{\begin{array}{cc}\rho -1,\hfill & \text{if}\mathit{\text{d}}\text{=0}\hfill \\ \rho -{\mathcal{D}}^{⁎}(\zeta +\varpi ),\hfill & \text{if}\mathit{\text{d}}\text{=1,2,…}\hfill \end{array}$$

It is self-evident that the underlying inequality is essential for ergodicity.

$$\rho <{\mathcal{D}}^{⁎}(\zeta +\varpi )$$

Humblett et al. [30] provide an easy way to assure non-ergodicity if the MC $\{A_n;n\in N\}$ satisfies Kaplan's criterion, notably, ${\beta }_d<\infty$ for all $d\ge 0$ and ∃ $d_0\in N$ s.t ${\beta }_d\ge 0$ for $d\ge d_0$. The fact that there is a w such that $(k_{vd})=0$ for $d<v-f$ and $v>0$, in which $(k_{vd})$ is the one step transition matrix of $\{A_n;n\in N\}$ implies that Kaplan's condition is fulfilled in our instance. In light of this, $\rho \ge {\mathcal{D}}^{⁎}(\zeta +\varpi )$ implies that the Markov Chain is non-ergodic. □

For the method $\{\mathrm{\Theta }(\epsilon ),\epsilon \ge 0\}$, we specify the probabilities ${\mathrm{\Pi }}_0(\epsilon )=P\{\mathrm{\Phi }(\epsilon )=0,\mathrm{\Theta }(\epsilon )=0\}$ and ${\mathcal{P}}_0(\epsilon )=P\{\mathrm{\Phi }(\epsilon )=1,\mathrm{\Theta }(\epsilon )=0\}$ and the prob. densities as follows,

$$\begin{gathered} {\mathrm{\Upsilon }}_n(\stackrel{`}{\upsilon },\epsilon )d\stackrel{`}{\upsilon }=P\{\mathrm{\Phi }(\epsilon )=1,\mathrm{\Theta }(\epsilon )=n,\stackrel{`}{\upsilon }\le {\mathcal{D}}^0(\epsilon )<\stackrel{`}{\upsilon }+d\stackrel{`}{\upsilon }\},\text{for}\epsilon ,\stackrel{`}{\upsilon }\ge 0\text{and}n\ge 1 \\ {\mathrm{\Gamma }}_{1,n}(\stackrel{`}{\upsilon },\epsilon )d\stackrel{`}{\upsilon }=P\{\mathrm{\Phi }(\epsilon )=2,\mathrm{\Theta }(\epsilon )=n,\stackrel{`}{\upsilon }\le {\mathcal{L}}_p^0(\epsilon )<\stackrel{`}{\upsilon }+d\stackrel{`}{\upsilon }\},\text{for}\epsilon ,\stackrel{`}{\upsilon },n\ge 0 \\ {\mathrm{\Gamma }}_{2,n}(\stackrel{`}{\upsilon },\tilde{\varsigma },\epsilon )d\stackrel{`}{\upsilon }=P\{\mathrm{\Phi }(\epsilon )=3,\mathrm{\Theta }(\epsilon )=n,\stackrel{`}{\upsilon }\le {\mathcal{L}}_p^0(\epsilon )<\stackrel{`}{\upsilon }+d\stackrel{`}{\upsilon },\tilde{\varsigma }\le {\mathcal{L}}_b^0(\epsilon )<\tilde{\varsigma }+d\tilde{\varsigma }\},\text{for}\epsilon ,(\stackrel{`}{\upsilon },\tilde{\varsigma }),n\ge 0 \\ {\mathrm{\Gamma }}_{b,n}(\tilde{\varsigma },\epsilon )d\stackrel{`}{\upsilon }=P\{\mathrm{\Phi }(\epsilon )=4,\mathrm{\Theta }(\epsilon )=n,\tilde{\varsigma }\le {\mathcal{L}}_b^0(\epsilon )<\tilde{\varsigma }+d\tilde{\varsigma }\},\text{for}\epsilon ,\tilde{\varsigma },n\ge 0 \\ {\mathrm{\Lambda }}_{v,n}(\stackrel{`}{\upsilon },\epsilon )d\stackrel{`}{\upsilon }=P\{\mathrm{\Phi }(\epsilon )=5,\mathrm{\Theta }(\epsilon )=n,\stackrel{`}{\upsilon }\le {\mathcal{L}}_v^0(\epsilon )<\stackrel{`}{\upsilon }+d\stackrel{`}{\upsilon }\},\text{for}\epsilon ,\stackrel{`}{\upsilon },n\ge 0 \end{gathered}$$

The sequel, in our opinion, meets the stability criteria, hence we can assign,

$${\mathrm{\Upsilon }}_n(\stackrel{`}{\upsilon })=li{m}_{\epsilon \to \infty }{\mathrm{\Upsilon }}_n(\stackrel{`}{\upsilon },\epsilon );{\mathrm{\Gamma }}_{1,n}(\stackrel{`}{\upsilon })=li{m}_{\epsilon \to \infty }{\mathrm{\Gamma }}_{1,n}(\stackrel{`}{\upsilon },\epsilon );{\mathrm{\Gamma }}_{2,n}(\stackrel{`}{\upsilon },\tilde{\varsigma })=li{m}_{\epsilon \to \infty }{\mathrm{\Gamma }}_{2,n}(\stackrel{`}{\upsilon },\tilde{\varsigma },\epsilon );{\mathrm{\Gamma }}_{b,n}(\stackrel{`}{\upsilon })=li{m}_{\epsilon \to \infty }{\mathrm{\Gamma }}_{b,n}(\stackrel{`}{\upsilon },\epsilon );{\mathrm{\Lambda }}_{i,n}(\stackrel{`}{\upsilon })=li{m}_{\epsilon \to \infty }{\mathrm{\Lambda }}_{i,n}(\stackrel{`}{\upsilon },\epsilon )$$

Using SVT, we build the below system of equations which regulate the dynamics of the system behaviour.

$$(\zeta +\varpi ){\mathcal{P}}_0=\gamma p{\mathrm{\Pi }}_0$$ (3.1)

$$(\zeta +\varpi +\gamma ){\mathrm{\Pi }}_0=\gamma q{\mathrm{\Pi }}_0+\underset{0}{\overset{\infty }{\int }}{\mathrm{\Gamma }}_{1,0}(\stackrel{`}{\upsilon }){\chi }_p(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }+\underset{0}{\overset{\infty }{\int }}{\mathrm{\Gamma }}_{b,0}(\tilde{\varsigma }){\chi }_p(\tilde{\varsigma })d\tilde{\varsigma }+\underset{0}{\overset{\infty }{\int }}{\mathrm{\Lambda }}_{v,0}(\stackrel{`}{\upsilon }){\chi }_v(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }$$ (3.2)

$$\frac{d{\mathrm{\Upsilon }}_n(\stackrel{`}{\upsilon })}{d\stackrel{`}{\upsilon }}=-(\zeta +\varpi +\phi (\stackrel{`}{\upsilon })){\mathrm{\Upsilon }}_n(\stackrel{`}{\upsilon });n\ge 0$$ (3.3)

$$\frac{d{\mathrm{\Gamma }}_{1,n}(\stackrel{`}{\upsilon })}{d\stackrel{`}{\upsilon }}+(\zeta +{\chi }_p(\stackrel{`}{\upsilon })){\mathrm{\Gamma }}_{1,n}(\stackrel{`}{\upsilon })=\zeta (1-b){\mathrm{\Gamma }}_{1,n}\stackrel{`}{\upsilon }+\zeta b\sum _{k=1}^n{\vartheta }_k{\mathrm{\Gamma }}_{1,n-k}(\stackrel{`}{\upsilon });n\ge 1$$ (3.4)

$$\frac{d{\mathrm{\Gamma }}_{2,n}(\stackrel{`}{\upsilon },\tilde{\varsigma })}{d\stackrel{`}{\upsilon }}+(\zeta +{\chi }_p(\stackrel{`}{\upsilon })){\mathrm{\Gamma }}_{2,n}(\stackrel{`}{\upsilon },\tilde{\varsigma })=\zeta (1-b){\mathrm{\Gamma }}_{2,n}(\stackrel{`}{\upsilon },\tilde{\varsigma })+\zeta b\sum _{k=1}^n{\vartheta }_k{\mathrm{\Gamma }}_{2,n-k}(\stackrel{`}{\upsilon },\tilde{\varsigma });n\ge 1$$ (3.5)

$$\frac{d{\mathrm{\Gamma }}_{b,n}(\tilde{\varsigma })}{d\tilde{\varsigma }}+(\zeta +\varpi {\chi }_b(\tilde{\varsigma })){\mathrm{\Gamma }}_{b,n}(\tilde{\varsigma })=\zeta (1-b){\mathrm{\Gamma }}_{b,n}(\tilde{\varsigma })+\zeta b\sum _{k=1}^n{\vartheta }_k{\mathrm{\Gamma }}_{b,n-k}(\tilde{\varsigma })+\underset{0}{\overset{\infty }{\int }}{\mathrm{\Gamma }}_{2,n}(\stackrel{`}{\upsilon },\tilde{\varsigma }){\chi }_p(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon };n\ge 1$$ (3.6)

$$\frac{d{\mathrm{\Lambda }}_{1,n}(\stackrel{`}{\upsilon })}{d\stackrel{`}{\upsilon }}+(\zeta +\gamma +{\chi }_v(\stackrel{`}{\upsilon })){\mathrm{\Lambda }}_{1,n}(\stackrel{`}{\upsilon })=\zeta (1-b){\mathrm{\Lambda }}_{v,n}\stackrel{`}{\upsilon }+\zeta b\sum _{k=1}^n{\vartheta }_k{\mathrm{\Lambda }}_{v,n-k}(\stackrel{`}{\upsilon });n\ge 1$$ (3.7)

Here are the SS boundary requirements at $\stackrel{`}{\upsilon }=0$, $\tilde{\varsigma }=0$

$${\mathrm{\Upsilon }}_n(0)=\underset{0}{\overset{\infty }{\int }}{\mathrm{\Gamma }}_{1,n}(\stackrel{`}{\upsilon }){\chi }_p(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }+\underset{0}{\overset{\infty }{\int }}{\mathrm{\Gamma }}_{b,n}(\tilde{\varsigma }){\chi }_b(\tilde{\varsigma })d\tilde{\varsigma }+\underset{0}{\overset{\infty }{\int }}{\mathrm{\Lambda }}_{v,n}(\stackrel{`}{\upsilon }){\chi }_v(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon };n\ge 1$$ (3.8)

$${\mathrm{\Gamma }}_{1,n}(0)=\varpi \underset{0}{\overset{\infty }{\int }}{\mathrm{\Upsilon }}_n(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon };n\ge 1$$ (3.9)

$${\mathrm{\Gamma }}_{2,n}(0,\tilde{\varsigma })=\varpi ({\mathrm{\Gamma }}_{b,n}(\tilde{\varsigma }));n\ge 0$$ (3.10)

$${\mathrm{\Gamma }}_{b,0}(0)=\underset{0}{\overset{\infty }{\int }}{\mathrm{\Upsilon }}_1(\stackrel{`}{\upsilon })\phi (\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }+\zeta (1-s)\underset{0}{\overset{\infty }{\int }}{\mathrm{\Upsilon }}_1(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }+\gamma \underset{0}{\overset{\infty }{\int }}{\mathrm{\Lambda }}_{v,0}(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon };n\ge 0$$ (3.11)

$${\mathrm{\Gamma }}_{b,n}(0)=\underset{0}{\overset{\infty }{\int }}{\mathrm{\Upsilon }}_{n+1}(\stackrel{`}{\upsilon })\phi (\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }+\zeta s\sum _{k=1}^n\underset{0}{\overset{\infty }{\int }}{\mathrm{\Upsilon }}_{n-k+1}(\stackrel{`}{\upsilon }){\vartheta }_{k}d\stackrel{`}{\upsilon }+\zeta (1-s)\underset{0}{\overset{\infty }{\int }}{\mathrm{\Upsilon }}_{n+1}(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }+\gamma \underset{0}{\overset{\infty }{\int }}{\mathrm{\Lambda }}_{v,n}(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon };n\ge 1$$ (3.12)

$${\mathrm{\Lambda }}_{v,n}(0)=\{\begin{array}{c}(\zeta +\varpi ){\mathrm{\Pi }}_0;n=0\hfill \\ 0;n\ge 1\hfill \end{array}$$ (3.13)

The normalizing condition is

$${\mathcal{P}}_0+{\mathrm{\Pi }}_0+\sum _{n=1}^{\infty }\underset{0}{\overset{\infty }{\int }}{\mathrm{\Upsilon }}_n(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }+\sum _{n=0}^{\infty }[\underset{0}{\overset{\infty }{\int }}{\mathrm{\Gamma }}_{1,n}(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }+\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{\mathrm{\Gamma }}_{2,n}(\stackrel{`}{\upsilon },\tilde{\varsigma })d\stackrel{`}{\upsilon }d\tilde{\varsigma }+\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{\mathrm{\Gamma }}_{b,n}(\tilde{\varsigma })d\tilde{\varsigma }+\underset{0}{\overset{\infty }{\int }}{\mathrm{\Lambda }}_{v,n}(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }]=1$$ (3.14)

3.3. The steady state solution

The SS solution to the RQ model is generated utilising generating function method. In order to solve the aforementioned equations, we define the GFs for $|\breve{\xi }|<1$ as follows:

$$\mathrm{\Upsilon }(\stackrel{`}{\upsilon },\breve{\xi })=\sum _{n=1}^{\infty }{\mathrm{\Upsilon }}_n(\stackrel{`}{\upsilon }){\breve{\xi }}^n;\mathrm{\Upsilon }(0,\breve{\xi })=\sum _{n=1}^{\infty }{\mathrm{\Upsilon }}_n(0){\breve{\xi }}^n;{\mathrm{\Gamma }}_1(\stackrel{`}{\upsilon },\breve{\xi })=\sum _{n=0}^{\infty }{\mathrm{\Gamma }}_{1,n}(\stackrel{`}{\upsilon }){\breve{\xi }}^n;{\mathrm{\Gamma }}_1(0,\breve{\xi })=\sum _{n=0}^{\infty }{\mathrm{\Gamma }}_{1,0}(0){\breve{\xi }}^n;{\mathrm{\Gamma }}_2(\stackrel{`}{\upsilon },\tilde{\varsigma },\breve{\xi })=\sum _{n=0}^{\infty }{\mathrm{\Gamma }}_{2,n}(\stackrel{`}{\upsilon },\widehat{)}{\breve{\xi }}^n;{\mathrm{\Gamma }}_2(0,\tilde{\varsigma },\breve{\xi })=\sum _{n=0}^{\infty }{\mathrm{\Gamma }}_{2,0}(0,\tilde{\varsigma }){\breve{\xi }}^n;{\mathrm{\Gamma }}_b(\tilde{\varsigma },\breve{\xi })=\sum _{n=0}^{\infty }{\mathrm{\Gamma }}_{b,n}(\tilde{\varsigma }){\breve{\xi }}^n;{\mathrm{\Gamma }}_b(0,\breve{\xi })=\sum _{n=0}^{\infty }{\mathrm{\Gamma }}_{b,0}(0){\breve{\xi }}^n;{\mathrm{\Lambda }}_v(\stackrel{`}{\upsilon },\breve{\xi })=\sum _{n=0}^{\infty }{\mathrm{\Lambda }}_{v,n}(\stackrel{`}{\upsilon }){\breve{\xi }}^n;{\mathrm{\Lambda }}_v(\stackrel{`}{\upsilon },\breve{\xi })=\sum _{n=0}^{\infty }{\mathrm{\Lambda }}_{v,0}(\stackrel{`}{\upsilon }){\breve{\xi }}^n;\mathcal{W}(\breve{\xi })=\sum _{n=1}^{\infty }{\vartheta }_n{\breve{\xi }}^n$$

By multiplying the SS equations and SS boundary requirements from (3.3) to (3.13) by ${\breve{\xi }}^n$ and summing over n, $(n=0,1,2,...)$

$$\frac{\partial }{\partial \stackrel{`}{\upsilon }}\mathrm{\Upsilon }(\stackrel{`}{\upsilon },\breve{\xi })+[\zeta +\varpi +\phi (\stackrel{`}{\upsilon })]\mathrm{\Upsilon }(\stackrel{`}{\upsilon },\breve{\xi })=0$$ (3.15)

$$\frac{\partial }{\partial \stackrel{`}{\upsilon }}{\mathrm{\Gamma }}_1(\stackrel{`}{\upsilon },\breve{\xi })+[\zeta b(1-\vartheta (\breve{\xi }))+{\chi }_p(\stackrel{`}{\upsilon })]{\mathrm{\Gamma }}_1(\stackrel{`}{\upsilon },\breve{\xi })=0$$ (3.16)

$$\frac{\partial }{\partial \stackrel{`}{\upsilon }}{\mathrm{\Gamma }}_2(\stackrel{`}{\upsilon },\tilde{\varsigma },\breve{\xi })+[\zeta b(1-\vartheta (\breve{\xi }))+{\chi }_p(\stackrel{`}{\upsilon })]{\mathrm{\Gamma }}_2(\stackrel{`}{\upsilon },\tilde{\varsigma },\breve{\xi })=0$$ (3.17)

$$\frac{\partial }{\partial \stackrel{`}{\upsilon }}{\mathrm{\Gamma }}_b(\tilde{\varsigma },\breve{\xi })+[\zeta b(1-\vartheta (\breve{\xi }))+\varpi +{\chi }_b(\stackrel{`}{\upsilon })]{\mathrm{\Gamma }}_b(\stackrel{`}{\upsilon },\tilde{\varsigma },\breve{\xi })=\underset{0}{\overset{\infty }{\int }}{\mathrm{\Gamma }}_2(\stackrel{`}{\upsilon },\tilde{\varsigma },\breve{\xi }){\chi }_p(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }$$ (3.18)

$$\frac{\partial }{\partial \stackrel{`}{\upsilon }}{\mathrm{\Lambda }}_v(\stackrel{`}{\upsilon },\breve{\xi })+[\zeta b(1-\vartheta (\breve{\xi }))+\gamma +{\chi }_v(\stackrel{`}{\upsilon })]{\mathrm{\Lambda }}_v(\stackrel{`}{\upsilon },\breve{\xi })=0$$ (3.19)

$$\mathrm{\Upsilon }(0,\breve{\xi })=\underset{0}{\overset{\infty }{\int }}{\mathrm{\Gamma }}_1(\stackrel{`}{\upsilon },\breve{\xi }){\chi }_p(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }+\underset{0}{\overset{\infty }{\int }}{\mathrm{\Gamma }}_b(\tilde{\varsigma },\breve{\xi }){\chi }_b(\tilde{\varsigma })d\tilde{\varsigma }+\underset{0}{\overset{\infty }{\int }}{\mathrm{\Lambda }}_v(\stackrel{`}{\upsilon },\breve{\xi }){\chi }_v(\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }-(\zeta +\varpi +\gamma p){\mathrm{\Pi }}_0$$ (3.20)

$${\mathrm{\Gamma }}_1(0,\breve{\xi })=\varpi \underset{0}{\overset{\infty }{\int }}\mathrm{\Upsilon }(\stackrel{`}{\upsilon },\breve{\xi })d\stackrel{`}{\upsilon }+\varpi {\mathcal{P}}_0$$ (3.21)

$${\mathrm{\Gamma }}_2(0,\tilde{\varsigma },\breve{\xi })=\varpi {\mathrm{\Gamma }}_b(\tilde{\varsigma },\breve{\xi })$$ (3.22)

$${\mathrm{\Gamma }}_b(0,\breve{\xi })=\frac{1}{\breve{\xi }}\underset{0}{\overset{\infty }{\int }}\mathrm{\Upsilon }(\stackrel{`}{\upsilon },\breve{\xi })\phi (\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }+\frac{\zeta s}{\breve{\xi }}\underset{0}{\overset{\infty }{\int }}\mathrm{\Upsilon }(\stackrel{`}{\upsilon },\breve{\xi })\vartheta (\stackrel{`}{\upsilon })d\stackrel{`}{\upsilon }+\zeta s{\mathcal{P}}_0+\frac{\zeta (1-s)}{\breve{\xi }}\underset{0}{\overset{\infty }{\int }}\mathrm{\Upsilon }(\stackrel{`}{\upsilon },\breve{\xi })d\stackrel{`}{\upsilon }+\gamma \underset{0}{\overset{\infty }{\int }}{\mathrm{\Lambda }}_v(\stackrel{`}{\upsilon },\breve{\xi })d\stackrel{`}{\upsilon }$$ (3.23)

$${\mathrm{\Lambda }}_v(0,\breve{\xi })=(\zeta +\varpi ){\mathrm{\Pi }}_0$$ (3.24)

Solving the partial differential eqns. (3.15) to (3.19), we obtain

$$\mathrm{\Upsilon }(\stackrel{`}{\upsilon },\breve{\xi })=\mathrm{\Upsilon }(0,\breve{\xi })[1-\mathcal{D}(\stackrel{`}{\upsilon })]e^{-(\zeta +\varpi )\stackrel{`}{\upsilon }}$$ (3.25)

$${\mathrm{\Gamma }}_1(\stackrel{`}{\upsilon },\breve{\xi })={\mathrm{\Gamma }}_1(0,\breve{\xi })[1-{\mathcal{L}}_p(\stackrel{`}{\upsilon })]e^{-{\mathcal{N}}_p(\breve{\xi })\stackrel{`}{\upsilon }}$$ (3.26)

$${\mathrm{\Gamma }}_2(\stackrel{`}{\upsilon },\tilde{\varsigma },\breve{\xi })={\mathrm{\Gamma }}_2(0,\tilde{\varsigma },\breve{\xi })[1-{\mathcal{L}}_p(\stackrel{`}{\upsilon })]e^{-{\mathcal{N}}_p(\breve{\xi })\stackrel{`}{\upsilon }}$$ (3.27)

$${\mathrm{\Gamma }}_b(\tilde{\varsigma },\breve{\xi })={\mathrm{\Gamma }}_b(0,\breve{\xi })[1-{\mathcal{L}}_b(\tilde{\varsigma })]e^{{\mathcal{N}}_b(\breve{\xi })\tilde{\varsigma }}$$ (3.28)

$${\mathrm{\Lambda }}_v(\stackrel{`}{\upsilon },\breve{\xi })={\mathrm{\Lambda }}_v(0,\breve{\xi })[1-{\mathcal{S}}_v(\stackrel{`}{\upsilon })]e^{-{\mathcal{N}}_v(\breve{\xi })\stackrel{`}{\upsilon }}$$ (3.29)

where ${\mathcal{N}}_p(\breve{\xi })=\zeta b(1-\vartheta (\breve{\xi }))$; ${\mathcal{N}}_b(\breve{\xi })=\zeta b(1-\vartheta (\breve{\xi }))+\varpi (1-{\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi })))$ and ${\mathcal{N}}_v(\breve{\xi })=\zeta b(1-\vartheta (\breve{\xi }))+\gamma$

Inserting the eqns. (3.25), (3.28) and (3.29) in (3.21) to (3.23) and after making some computations, we eventually arrive at,

$${\mathrm{\Gamma }}_1(0,\breve{\xi })=\varpi \mathrm{\Upsilon }(0,\breve{\xi }){\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )+\varpi {\mathcal{P}}_0$$ (3.30)

$${\mathrm{\Gamma }}_b(0,\breve{\xi })=\frac{\mathrm{\Upsilon }(0,\breve{\xi })}{\breve{\xi }}[{\mathcal{D}}^{⁎}(\zeta +\varpi )+\breve{\xi }\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )[1-s+s\vartheta (\breve{\xi })]]+\zeta s{\mathcal{P}}_0+(\zeta +\varpi ){\mathrm{\Pi }}_{0}G(\breve{\xi })$$ (3.31)

$${\mathrm{\Gamma }}_2(0,\tilde{\varsigma },\breve{\xi })=\varpi {\mathrm{\Gamma }}_b(0,\breve{\xi })[1-{\mathcal{L}}_b(\tilde{\varsigma })]e^{{\mathcal{N}}_b(\breve{\xi })\tilde{\varsigma }}$$ (3.32)

where

$${\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )=\frac{1-{\mathcal{D}}^{⁎}(\zeta +\varpi )}{(\zeta +\varpi )}andG(\breve{\xi })=\frac{\gamma [1-{\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))]}{{\mathcal{N}}_p(\breve{\xi })+\gamma }$$

By substituting the eqns. (3.26) to (3.29) in (3.20), we get,

$$\mathrm{\Upsilon }(0,\breve{\xi })={\mathrm{\Gamma }}_1(0,\breve{\xi }){\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi }))+{\mathrm{\Gamma }}_b(0,\breve{\xi }){\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi }))+{\mathrm{\Lambda }}_v(0,\breve{\xi }){\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))-(\zeta +\varpi +\gamma p){\mathrm{\Pi }}_0$$ (3.33)

Corollary 3.2

The marginal probability dist. of the no of patients in the orbit when the server is idle, active serving a priority request without pre-empting a regular patient, engaged serving a priority request while pre-empting a regular patient, active serving an regular patient request, and on working vacation are stated below under the stability condition $\rho <{\mathcal{D}}^{⁎}(\zeta +\varpi )$

$$\mathrm{\Upsilon }(\breve{\xi })=\frac{\breve{\xi }{\mathrm{\Pi }}_0{\mathcal{D}}^{⁎}(\zeta +\varpi )\{(\zeta +\varpi )[{\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))+{\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi }))G(\breve{\xi })-1]-(\frac{\gamma p}{\zeta +\varpi })[\varpi (1-{\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi })))+\zeta (1-s{\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi })))]\}}{Dr(\breve{\xi })}$$ (3.34)

$${\mathrm{\Gamma }}_1(\breve{\xi })=\frac{\begin{array}{c}\hfill \varpi {\mathrm{\Pi }}_0\{\breve{\xi }[1-{\mathcal{D}}^{⁎}(\zeta +\varpi )][{\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))+{\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi }))G(\breve{\xi })-1]+\left(\frac{\gamma p}{\zeta +\varpi }\right){\mathcal{D}}^{⁎}(\zeta +\varpi )\hfill \\ \hfill [\breve{\xi }[1-\zeta {\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi }))[1+s\vartheta (\breve{\xi })]]-{\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi }))]\}[1-{\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi }))]\hfill \end{array}}{{\mathcal{N}}_p(\breve{\xi })Dr(\breve{\xi })}$$ (3.35)

$${\mathrm{\Gamma }}_2(\breve{\xi })=\frac{\begin{array}{c}\hfill \varpi {\mathrm{\Pi }}_0\{((\zeta +\varpi )({\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))-1)-\gamma p)({\mathcal{D}}^{⁎}(\zeta +\varpi )+\breve{\xi }\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1-s+s\vartheta (\breve{\xi })))\hfill \\ \hfill +\left(\frac{\gamma p}{\zeta +\varpi }\right)(\zeta s\breve{\xi }+\varpi {\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi }))({\mathcal{D}}^{⁎}(\zeta +\varpi )+\breve{\xi }\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1-s+s\vartheta (\breve{\xi }))))\hfill \\ \hfill +\breve{\xi }(\zeta +\varpi )G(\breve{\xi })(1-\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ){\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))\}[1-{\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi }))][1-{\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi }))]\hfill \end{array}}{{\mathcal{A}}_p(\breve{\xi }){\mathcal{A}}_b(\breve{\xi })Dr(\breve{\xi })}$$ (3.36)

$${\mathrm{\Gamma }}_b(\breve{\xi })=\frac{\begin{array}{c}\hfill {\mathrm{\Pi }}_0\{((\zeta +\varpi )({\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))-1)-\gamma p)({\mathcal{D}}^{⁎}(\zeta +\varpi )+\breve{\xi }\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1-s+s\vartheta (\breve{\xi })))\hfill \\ \hfill +\left(\frac{\gamma p}{\zeta +\varpi }\right)(\zeta s\breve{\xi }+\varpi {\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi }))({\mathcal{D}}^{⁎}(\zeta +\varpi )+\breve{\xi }\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1-s+s\vartheta (\breve{\xi }))))\hfill \\ \hfill +\breve{\xi }(\zeta +\varpi )G(\breve{\xi })(1-\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ){\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))\}[1-{\mathcal{L}}_b^{⁎}{\mathcal{N}}_b(\breve{\xi })]\hfill \end{array}}{Dr(\breve{\xi }){\mathcal{A}}_b(\breve{\xi })}$$ (3.37)

$${\mathrm{\Lambda }}_v(\breve{\xi })=\frac{(\zeta +\varpi ){\mathrm{\Pi }}_{0}G(\breve{\xi })}{\gamma }$$ (3.38)

where,

$${\mathrm{\Pi }}_0=\frac{{\mathcal{D}}^{⁎}(\zeta +\varpi )-\rho }{\sigma }$$ (3.39)

$${\mathcal{P}}_0=\frac{\gamma p{\mathcal{D}}^{⁎}(\zeta +\varpi )-\rho }{(\zeta +\varpi )\sigma }$$ (3.40)

$$\begin{gathered} \sigma =(\frac{(\zeta +\varpi )(1-{\mathcal{L}}_v^{⁎}(\gamma ))}{\gamma }+1)({\mathcal{D}}^{⁎}(\zeta +\varpi )-\rho )+(\zeta +\varpi )(1-{\mathcal{L}}_v^{⁎}(\gamma )){\mu }_b^{(1)}\{\zeta b{\mathcal{I}}_1[\frac{1}{\gamma }-\varpi \\ {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(\frac{1}{\gamma }-{\mu }_p^{(1)})-\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )]+{\mathcal{D}}^{⁎}(\zeta +\varpi )\}[1+\varpi {\mu }_p^{(1)}]+(1-{\mathcal{L}}_v^{⁎}(\gamma ))(\frac{1}{\gamma }+{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}])\zeta b{\mathcal{I}}_1(1-{\mathcal{D}}^{⁎}(\zeta +\varpi ))[1+\varpi {\mu }_p^{(1)}]+(\frac{\gamma p}{\zeta +\varpi })\{[\zeta s+\zeta b{\mathcal{I}}_1\varpi {\mathcal{D}}^{⁎}(\zeta +\varpi ){\mu }_b^{(1)}[1-\zeta (1+s)]+\zeta \varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1+s+s{\mathcal{I}}_1)]{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}]+{\mu }_p^{(1)}\varpi {\mathcal{D}}^{⁎}(\zeta +\varpi )[1-\zeta (1+s)(1+\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}])-\zeta s{\mathcal{I}}_1]\}; \end{gathered}$$

and

$$\rho =({\mathcal{D}}^{⁎}(\zeta +\varpi )+\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}]+\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))\zeta b{\mathcal{I}}_1{\mu }_p^{(1)}-\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )s{\mathcal{I}}_1$$

Here, the below mentioned normalising constraint might be utilized to estimate the unknowns, ${\mathcal{P}}_0$ and ${\mathrm{\Pi }}_0$ the chance that the server will be inactive during a regular busy phase and the chance that the server is inactive in WV phase while there are zero patients in the orbit. With this in mind, by setting $\breve{\xi }=1$ in eqns. (3.34) - (3.38) and employing the rule of L'Hospitals where necessary, one may get

$${\mathcal{P}}_0+{\mathrm{\Pi }}_0+\mathrm{\Upsilon }(1)+{\mathrm{\Gamma }}_1(1)+{\mathrm{\Gamma }}_2(1)+{\mathrm{\Gamma }}_b(1)+{\mathrm{\Lambda }}_v(1)=1.$$

Theorem 3.3

In the presence of a stability constraint $\rho <{\mathcal{D}}^{⁎}(\zeta +\varpi )$ , the PGF of the no. of patients in the system $K_e(\breve{\xi })$ and the orbit $R_e(\breve{\xi })$ size dist. at a fixed time are computed as follows:

$$K_e(\breve{\xi })=\frac{N{r}_e(\breve{\xi })}{Dr(\breve{\xi }){\mathcal{N}}_p(\breve{\xi })}$$ (3.41)

$$\begin{gathered} N{r}_e(\breve{\xi })={\mathrm{\Pi }}_0\times \{[1+\left(\frac{\gamma p}{\zeta +\varpi }\right)+\frac{(\zeta +\varpi )\breve{\xi }}{\gamma }G(\breve{\xi })]Dr(\breve{\xi }){\mathcal{N}}_p(\breve{\xi })+\breve{\xi }[1-{\mathcal{D}}^{⁎}(\zeta +\varpi )][{\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))+{\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi }))G(\breve{\xi })-1]({\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi }))+\breve{\xi }\varpi (1-{\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi }))))+\breve{\xi }(1-{\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi })))\{((\zeta +\varpi ) \\ ({\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))-1)-\gamma p)({\mathcal{D}}^{⁎}(\zeta +\varpi )+\breve{\xi }\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1-s+s\vartheta (\breve{\xi })))+\left(\frac{\gamma p}{\zeta +\varpi }\right) \\ (\zeta s\breve{\xi }+\varpi {\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi }))({\mathcal{D}}^{⁎}(\zeta +\varpi )+\breve{\xi }\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1+s\vartheta (\breve{\xi }))))+\breve{\xi }(\zeta +\varpi )G(\breve{\xi })(1-\varpi ){\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ){\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))\}+\breve{\xi }\left(\frac{\gamma p}{\zeta +\varpi }\right)\{\varpi (1-{\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi })))[{\mathcal{D}}^{⁎}(\zeta +\varpi )[\breve{\xi }(1-\zeta {\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi })) \\ (1+s\vartheta (\breve{\xi })))-{\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi }))]-{\mathcal{N}}_p(\breve{\xi }){\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )]-{\mathcal{N}}_p(\breve{\xi })\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1-s{\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi })))\}\}Dr(\breve{\xi })=\breve{\xi }-\breve{\xi }\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ){\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi }))-[{\mathcal{D}}^{⁎}(\zeta +\varpi )+\breve{\xi }\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )[1-s+s\vartheta (\breve{\xi })]]{\mathcal{N}}_p(\breve{\xi })=\zeta b(1-\vartheta (\breve{\xi })) \end{gathered}$$

$$R_e(\breve{\xi })=\frac{N{r}_u(\breve{\xi })}{Dr(\breve{\xi }){\mathcal{N}}_p(\breve{\xi })}$$ (3.42)

$$\begin{gathered} N{r}_e(\breve{\xi })={\mathrm{\Pi }}_0\times \{[1+\left(\frac{\gamma p}{\zeta +\varpi }\right)+\frac{(\zeta +\varpi )}{\gamma }G(\breve{\xi })]Dr(\breve{\xi }){\mathcal{N}}_p(\breve{\xi })+\breve{\xi }[1-{\mathcal{D}}^{⁎}(\zeta +\varpi )]{\mathcal{N}}_b(\breve{\xi })[{\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))+{\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi }))G(\breve{\xi })-1]+(1-{\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi })))\{((\zeta +\varpi )({\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))-1)-\gamma p) \\ ({\mathcal{D}}^{⁎}(\zeta +\varpi )+\breve{\xi }\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1-s+s\vartheta (\breve{\xi })))+\left(\frac{\gamma p}{\zeta +\varpi }\right)(\zeta s\breve{\xi }+\varpi {\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi })) \\ ({\mathcal{D}}^{⁎}(\zeta +\varpi )+\breve{\xi }\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1+s\vartheta (\breve{\xi }))))+\breve{\xi }(\zeta +\varpi )G(\breve{\xi })(1-\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ){\mathcal{L}}_v^{⁎}({\mathcal{N}}_v(\breve{\xi }))\}+\left(\frac{\gamma p}{\zeta +\varpi }\right)\{\varpi (1-{\mathcal{L}}_p^{⁎}({\mathcal{N}}_p(\breve{\xi })))[{\mathcal{D}}^{⁎}(\zeta +\varpi )[\breve{\xi }(1-\zeta {\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi }))(1+s\vartheta (\breve{\xi })))-{\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi }))] \\ -\breve{\xi }{\mathcal{N}}_p(\breve{\xi }){\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )]-\breve{\xi }{\mathcal{N}}_p(\breve{\xi })\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1-s{\mathcal{L}}_b^{⁎}({\mathcal{N}}_b(\breve{\xi })))\}\} \end{gathered}$$

where eqn. (3.39) represents ${\mathrm{\Pi }}_0$

Proof

The PGF of the no.of patients in the system $(K_e(\breve{\xi }))$ and in the orbit $(R_e(\breve{\xi }))$ have been derived with the aid of the below mentioned eqns.

$$K_e(\breve{\xi })={\mathcal{P}}_0+{\mathrm{\Pi }}_0+\mathrm{\Upsilon }(\breve{\xi })+\breve{\xi }\{{\mathrm{\Gamma }}_1(\breve{\xi })+{\mathrm{\Gamma }}_2(\breve{\xi })+{\mathrm{\Gamma }}_b(\breve{\xi })+{\mathrm{\Lambda }}_v(\breve{\xi })\}$$

and

$$R_e(\breve{\xi })={\mathcal{P}}_0+{\mathrm{\Pi }}_0+\mathrm{\Upsilon }(\breve{\xi })+\{{\mathrm{\Gamma }}_1(\breve{\xi })+{\mathrm{\Gamma }}_2(\breve{\xi })+{\mathrm{\Gamma }}_b(\breve{\xi })+{\mathrm{\Lambda }}_v(\breve{\xi })\}.$$

The eqns. (3.41) and (3.42) can be readily approximated by substituting the values from the eqns. (3.34) to (3.38) into aforementioned eqns. □

4. System performance measures

The SS probability while the server isn't in usage yet accessible within the system is represented by eqns. (3.39) and (3.40). As a result, the server state probabilities are obtained from eqns. (3.34) to (3.38), which are states below, when the system satisfies the stability criteria $\rho <{\mathcal{D}}^{⁎}(\zeta +\varpi )$.

• 1.
  Let ϒ denote the SS prob. that the server being inactive during the retrial time

  $$\mathrm{\Upsilon }=li{m}_{\breve{\xi }\to 1}\mathrm{\Upsilon }(\breve{\xi })={\mathrm{\Pi }}_0{\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )\zeta b{\mathcal{I}}_1\times \left\{\frac{(\zeta +\varpi )(1-{\mathcal{L}}_v^{⁎}(\gamma ))[\frac{1}{\gamma }+{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}]]+(\frac{\gamma p}{\zeta +\varpi })[\varpi {\mu }_p^{(1)}+\zeta s{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}]]}{{\mathcal{D}}^{⁎}(\zeta +\varpi )-\rho }\right\}$$
• 2.
  Let ${\mathrm{\Gamma }}_1$ denote the SS prob. where server being engaged serving a priority request without pre-empting a regular patient request,

  $${\mathrm{\Gamma }}_1=li{m}_{\breve{\xi }\to 1}{\mathrm{\Gamma }}_1(\breve{\xi })=\varpi {\mathrm{\Pi }}_0{\mu }_p^{(1)}\times \left\{\frac{\begin{array}{c}\hfill (1-{\mathcal{D}}^{⁎}(\zeta +\varpi ))(1-{\mathcal{L}}_v^{⁎}(\gamma ))\zeta b{\mathcal{I}}_1[\frac{1}{\gamma }+{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}]]+(\frac{\gamma p}{\zeta +\varpi })\hfill \\ \hfill {\mathcal{D}}^{⁎}(\zeta +\varpi )[1-\zeta (1+s)[1+\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}]]-\zeta s{\mathcal{I}}_1]\hfill \end{array}}{{\mathcal{D}}^{⁎}(\zeta +\varpi )-\rho }\right\}$$
• 3.
  Let ${\mathrm{\Gamma }}_2$ denote the SS prob. where server being engaged serving a priority request with pre-empting a regular patient request,

  $${\mathrm{\Gamma }}_2=li{m}_{\breve{\xi }\to 1}{\mathrm{\Gamma }}_2(\breve{\xi })=\varpi {\mathrm{\Pi }}_0{\mu }_b^{(1)}{\mu }_p^{(1)}\times \left\{\frac{\begin{array}{c}\hfill (\zeta +\varpi )(1-{\mathcal{L}}_v^{⁎}(\gamma ))[\zeta b{\mathcal{I}}_1[\frac{1}{\gamma }-\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(\frac{1}{\gamma }-{\mu }_p^{(1)})\hfill \\ \hfill -\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )]+{\mathcal{D}}^{⁎}(\zeta +\varpi )]+(\frac{\gamma p}{\zeta +\varpi })[\zeta s+\zeta b{\mathcal{I}}_1{\mu }_p^{(1)}\hfill \\ \hfill \varpi {\mathcal{D}}^{⁎}(\zeta +\varpi )[1-\zeta (1+s)]+\zeta \varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1+s+s{\mathcal{I}}_1)]\hfill \end{array}}{{\mathcal{D}}^{⁎}(\zeta +\varpi )-\rho }\right\}$$
• 4.
  Let ${\mathrm{\Gamma }}_b$ denote the SS prob. where server being engaged serving regular patient request,

  $${\mathrm{\Gamma }}_b=li{m}_{\breve{\xi }\to 1}{\mathrm{\Gamma }}_b(\breve{\xi })={\mathrm{\Pi }}_0{\mu }_b^{(1)}\times \left\{\frac{\begin{array}{c}\hfill (\zeta +\varpi )(1-{\mathcal{L}}_v^{⁎}(\gamma ))[\zeta b{\mathcal{I}}_1[\frac{1}{\gamma }-\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(\frac{1}{\gamma }-{\mu }_p^{(1)})\hfill \\ \hfill -\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )]+{\mathcal{D}}^{⁎}(\zeta +\varpi )]+(\frac{\gamma p}{\zeta +\varpi })[\zeta s+\zeta b{\mathcal{I}}_1{\mu }_p^{(1)}\hfill \\ \hfill \varpi {\mathcal{D}}^{⁎}(\zeta +\varpi )[1-\zeta (1+s)]+\zeta \varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1+s+s{\mathcal{I}}_1)]\hfill \end{array}}{{\mathcal{D}}^{⁎}(\zeta +\varpi )-\rho }\right\}$$
• 5.
  Let ${\mathrm{\Lambda }}_v$ denote the SS prob. where server is under WV

  $${\mathrm{\Lambda }}_v=li{m}_{\breve{\xi }\to 1}{\mathrm{\Lambda }}_v(\breve{\xi })=\frac{(\zeta +\varpi ){\mathrm{\Pi }}_0}{\gamma }(1-{\mathcal{L}}_v^{⁎}(\gamma ))$$

4.1. Average system size and orbit size

Upon entering SS, the average number of patients in the orbit $(L_q)$ is determined by differentiating (3.42) in relation to $\breve{\xi }$ and giving $\breve{\xi }=1$ under the stability condition.

$$L_q=R_e^{\prime }(1)=\underset{\breve{\xi }\to 1}{\lim }\frac{d}{d\breve{\xi }}{R}_e(\breve{\xi })={\mathrm{\Pi }}_0\left[\frac{N{r}_q^{‴}(1)D{r}_q^{″}(1)-D{r}_q^{‴}(1)N{r}_q^{″}(1)}{3{(D{r}_q^{″}(1))}^2}\right]$$ (4.43)

$$\begin{gathered} D{r}_q^{″}(1)=-2\zeta b{\mathcal{I}}_1[{\mathcal{D}}^{⁎}(\zeta +\varpi )-\rho ]D{r}_q^{‴}(1)=3\zeta b{\mathcal{I}}_1\{{\mathcal{D}}^{⁎}(\zeta +\varpi )+\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )\eta +\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )[2\zeta b{\mathcal{I}}_1{\mu }_p^{(1)}-\zeta b\kappa ]+2\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}][\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1+s{\mathcal{I}}_1)]+\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )s(2{\mathcal{I}}_1+{\mathcal{I}}_2)\}N{r}_q^{″}(1)=2(\zeta +\varpi )(1-{\mathcal{L}}_v^{⁎}(\gamma ))[\zeta b{\mathcal{I}}_1{\mu }_p^{(1)}[{\mathcal{D}}^{⁎}(\zeta +\varpi )+\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )\zeta b{\mathcal{I}}_1{\mu }_p^{(1)}-\zeta \\ {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )s{\mathcal{I}}_1]-\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}][\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1+s{\mathcal{I}}_1)]+1-\varpi \\ {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )]+2(\frac{\gamma p}{\zeta +\varpi })[\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}][\zeta s+\varpi \zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1+s{\mathcal{I}}_1)+\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )s\varpi \zeta b{\mathcal{I}}_1{\mu }_p^{(1)}[{\mathcal{D}}^{⁎}(\zeta +\varpi )\zeta (1+s)+{\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )]]]-2(1-{\mathcal{D}}^{⁎}(\zeta +\varpi ))[{\mathcal{L}}_v^{{⁎}^{\prime }}(\gamma )+G^{\prime }(1)-\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}]][\zeta b{\mathcal{I}}_1(1+\varpi {\mu }_p^{(1)})] \end{gathered}$$

$$\begin{gathered} N{r}_q^{‴}(1)=6[\frac{\zeta +\varpi }{\gamma }{G}^{\prime }(1)\zeta b{\mathcal{I}}_1{\mu }_p^{(1)}\rho ]+3[1+\left(\frac{\gamma p}{\zeta +\varpi }\right)+\frac{(\zeta +\varpi )}{\gamma }(1-{\mathcal{L}}_v^{⁎}(\gamma ))]\{\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}[({\mathcal{D}}^{⁎}(\zeta +\varpi )+\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))\zeta b+\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )\zeta b\kappa -\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )s{\mathcal{I}}_1-\zeta b\kappa \rho ]\}+3\{\zeta b\eta ((\zeta +\varpi ){\mathcal{L}}_v^{{⁎}^{\prime }}(\gamma ))({\mathcal{D}}^{⁎}(\zeta +\varpi )+\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))+(\zeta +\varpi )(1-{\mathcal{L}}_v^{⁎}(\gamma ))[-\zeta s{\mathcal{I}}_1{\mathcal{D}}^{⁎}(\zeta +\varpi ){\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )+\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}]+(\zeta +\varpi )G^{\prime }(1)(1-\varpi \\ {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))+\left(\frac{\gamma p}{\zeta +\varpi }\right)[\zeta s+\varpi [-\zeta b{\mathcal{I}}_1{\mu }_p^{(1)}({\mathcal{D}}^{⁎}(\zeta +\varpi )+\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))+\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ) \\ (1+s{\mathcal{I}}_1)]]-\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}][(\zeta +\varpi ){\mathcal{L}}_v^{{⁎}^{″}}(\gamma )({\mathcal{D}}^{⁎}(\zeta +\varpi )+\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))+\zeta s{\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ){\mathcal{I}}_2[(\zeta +\varpi )({\mathcal{L}}_v^{⁎}(\gamma )-1)-\gamma p]+(\zeta +\varpi )[(1-{\mathcal{L}}_v^{⁎}(\gamma ))\varpi \zeta b \\ {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )\kappa +\varpi \zeta b{\mathcal{I}}_1{\mu }_b^{(1)}{\mathcal{L}}_v^{{⁎}^{\prime }}(\gamma )]+(\frac{\gamma p}{\zeta +\varpi })\left[\varpi [\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )s{\mathcal{I}}_2-\zeta b\kappa ]\right]]\}+(1-{\mathcal{D}}^{⁎}(\zeta +\varpi ))\{6[\zeta b{\mathcal{I}}_1[1+\varpi {\mu }_p^{(1)}]({\mathcal{L}}_v^{{⁎}^{\prime }}(\gamma )+G^{\prime }(1)-\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}])]+3[\zeta b{\mathcal{I}}_1[1+\varpi {\mu }_p^{(1)}]({\mathcal{L}}_v^{{⁎}^{″}}(\gamma )+G^{″}(1)-\zeta b\eta )+({\mathcal{L}}_v^{{⁎}^{\prime }}(\gamma )+G^{\prime }(1)-\zeta b{\mathcal{I}}_1{\mu }_b^{(1)} \\ [1+\varpi {\mu }_p^{(1)}])\zeta b({\mathcal{I}}_2+\varpi \kappa )]\}+3(\frac{\gamma p}{\zeta +\varpi })[\zeta b{\mathcal{I}}_1{\mu }_p^{(1)}s\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )({\mathcal{D}}^{⁎}(\zeta +\varpi )\zeta (1+s)-{\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))[\zeta b\kappa [1+\varpi {\mu }_p^{(1)}]+\zeta b\eta ]] \end{gathered}$$

(ii)The average amount of patients in the orbit $(L_s)$ is determined by differentiating (3.41) with regard to $\breve{\xi }$ and giving $\breve{\xi }=1$ within the stability condition.

$$L_s=K_e^{\prime }(1)=\underset{\breve{\xi }\to 1}{\lim }\frac{d}{d\breve{\xi }}{K}_e(\breve{\xi })={\mathrm{\Pi }}_0\left[\frac{N{r}_s^{‴}(1)D{r}_q^{″}(1)-D{r}_q^{‴}(1)N{r}_q^{″}(1)}{3{(D{r}_q^{″}(1))}^2}\right]$$ (4.44)

$$\begin{gathered} N{r}_s^{‴}(1)=N{r}_q^{‴}(1)-6\{\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}]((\zeta +\varpi ){\mathcal{L}}_v^{{⁎}^{\prime }}(\gamma ))({\mathcal{D}}^{⁎}(\zeta +\varpi )+\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))+(\zeta +\varpi )(1-{\mathcal{L}}_v^{⁎}(\gamma ))[-\zeta s{\mathcal{I}}_1{\mathcal{D}}^{⁎}(\zeta +\varpi ){\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )+\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )\zeta b{\mathcal{I}}_1{\mu }_b^{(1)}]+(\zeta +\varpi )G^{\prime }(1)(1-\varpi {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))+(\frac{\gamma p}{\zeta +\varpi })[\zeta s+\varpi [-\zeta b{\mathcal{I}}_1{\mu }_p^{(1)}({\mathcal{D}}^{⁎}(\zeta +\varpi ) \\ +\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))+\zeta {\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi )(1+s{\mathcal{I}}_1)]]+(\frac{\gamma p}{\zeta +\varpi })[\varpi \zeta b{\mathcal{I}}_1{\mu }_b^{(1)}[1+\varpi {\mu }_p^{(1)}]s\zeta \\ b{\mathcal{I}}_1{\mu }_p^{(1)}({\mathcal{D}}^{⁎}(\zeta +\varpi )\zeta (1+s)-{\bar{\mathcal{D}}}^{⁎}(\zeta +\varpi ))]\} \end{gathered}$$

where,

$$\kappa ={\mathcal{I}}_2{\mu }_p^{(1)}-{\mathcal{I}}_1\zeta b{\mu }_p^{(2)};{\mathcal{L}}_v^{{⁎}^{\prime }}(\gamma )=\underset{0}{\overset{\infty }{\int }}\stackrel{`}{\upsilon }{e}^{-\gamma \stackrel{`}{\upsilon }}d{\mathcal{L}}_v(\stackrel{`}{\upsilon });\eta =\zeta b[{({\mathcal{I}}_2)}^2{\mu }_b^{(2)}{(1+\varpi {\mu }_p^{(1)})}^2+{\mathcal{I}}_2{\mu }_b^{(1)}(1+\varpi {\mu }_p^{(1)})+\varpi {({\mathcal{I}}_1)}^2{\mu }_b^{(1)}{\mu }_p^{(2)}];G^{\prime }(1)=\frac{\zeta }{\gamma }(1-{\mathcal{L}}_v^{⁎}(\gamma )+\gamma {\mathcal{L}}_v^{{⁎}^{\prime }}(\gamma ));G^{″}(1)={(\frac{\zeta }{\gamma })}^2[2(1-{\mathcal{L}}_v^{⁎}(\gamma )+\gamma {\mathcal{L}}_v^{{⁎}^{\prime }}(\gamma ))-{(\gamma )}^2{\mathcal{L}}_v^{{⁎}^{″}}(\gamma )]$$

(iii) The optimum amount of time a patient might remain in the system ($W_s$) and the optimum amount of time they might remain in the queue ($W_q$) are predicted by incorporating the Little's law. (i.e) $W_s=\frac{L_s}{\zeta }$ and $W_q=\frac{L_q}{\zeta }$,

5. Numerical examples

MATLAB has been incorporated into this section to illustrate the spectrum of potential outcomes for the system's dynamic response. Furthermore, regular service for both priority and regular patients, exponentially distributed retrials, and WV have been addressed. To ensure stability, a random selection is made among the numerical measurements. The computed values of distinct characteristics of the framework under consideration such as, the probabilities that the server is idle ${\mathrm{\Pi }}_0$ and ${\mathcal{P}}_0$, the average queue size $(L_q)$, average system size $(L_q)$, prob. that the server is engaged serving a priority patient without pre-empting an regular patient ${\mathrm{\Gamma }}_1$, prob. that the server is under WV have been presented in Table 1, Table 2, Table 3, Table 4, Table 5, Table 6 and it is depicted via 2D graphs in Fig. 4

Table 1.

The impact of arrival rate (ϖ) on ${\mathcal{P}}_0$, L_q, Γ₁, ϒ.

Arrival rate (ϖ)	${\mathcal{P}}_0$	Lq	Γ1	ϒ
1.0	1.1500	1.2259	0.0178	0.0427
1.5	1.0108	2.3162	0.0273	0.0621
2.0	0.9578	3.6647	0.0371	0.0802
2.5	0.9311	5.2149	0.0466	0.0960
3.0	0.9156	6.9309	0.0557	0.1093

Table 2.

The influence of priority arrival rate (ζ) on ${\mathcal{P}}_0$, Π₀, L_q, ϒ.

Priority Arrival rate (ζ)	${\mathcal{P}}_0$	Π0	Lq	ϒ
1.0	0.0721	0.1576	2.5551	0.0383
2.0	0.0643	0.1530	4.6619	0.0549
3.0	0.0600	0.1437	5.4775	0.0778
4.0	0.0594	0.1249	5.4776	0.1061
5.0	0.0356	0.0851	5.0293	0.1232

Table 3.

The influence of retrial rate (φ) on Π₀, L_q, Γ₂, Γ_b.

Retrial Rate (φ)	Π0	Lq	Γ2	Γb
1.0	0.1353	1.2743	0.0259	0.0863
2.0	0.0850	0.7566	0.0245	0.0817
3.0	0.0702	0.4022	0.0185	0.0615
4.0	0.0631	0.2136	0.0136	0.0454
5.0	0.0590	0.1102	0.0099	0.0331

Table 4.

The influence of lower service rate (χ_v) on L_s, L_q, ϒ, Π₀.

Lower Service rate (χv)	Ls	Lq	ϒ	Π0
6.0	3.0841	1.7127	0.2112	0.5142
6.5	2.8143	1.3213	0.2250	0.4940
7.0	2.6544	1.2109	0.2357	0.4783
7.5	2.5670	0.9834	0.2442	0.4658
8.0	2.5304	0.7842	0.2511	0.4766

Table 5.

The influence of balking probability (b) on Π₀, W_q, ϒ, Γ₁.

Balking probability (b)	Π0	Wq	ϒ	Γ1
0.81	0.2738	0.3887	0.0564	0.0757
0.82	0.2741	0.3141	0.0567	0.0760
0.83	0.2745	0.2390	0.0569	0.0762
0.84	0.2748	0.1634	0.0571	0.0764
0.85	0.2752	0.0871	0.0573	0.0766

Table 6.

The influence of reneging probability (s) on Π₀, W_q, W_s, Γ_b.

Reneging probability (s)	Π0	Wq	Ws	Γb
0.1	0.3337	3.0111	2.4384	0.0050
0.2	0.3404	2.6468	1.6003	0.0366
0.3	0.3472	2.2720	0.9866	0.0688
0.4	0.3541	1.8867	0.7185	0.1018
0.5	0.3612	1.4906	0.5684	0.1355

Figure 4.

2D representation of effects of various parameters.

Table 1 clearly displays that as arrival rate (ζ) mounts, $L_q$, ${\mathrm{\Gamma }}_1$ and ϒ also rises, whereas ${\mathcal{P}}_0$ subsides for the value of ${\mathcal{I}}_1$=0.33, b=0.91, ϖ=6.5, γ=2.5, s=0.095, p= 0.01, ${\mathcal{D}}^{⁎}(\zeta +\varpi )$=4.5, ${\mu }_p^{(1)}$ = 0.37 and ${\mu }_p^{(2)}$ = 0.137. Further, Fig. 4(a) displays as the arrival rate rises, $L_q$ and ϒ also elevates, however, ${\mathcal{P}}_0$, decline. As the arrival rate (ζ) increases, the software must dynamically allocate more resources to handle the growing queue length $(L_q)$ and maintain service efficiency. With higher service $({\mathrm{\Gamma }}_1)$ and retrial (ϒ) rates, there's increased demand for services, necessitating robust scheduling algorithms to manage the load without compromising quality. The decreasing probability of server inactivity $({\mathcal{P}}_0)$ indicates higher system utilization, underscoring the need for robust monitoring tools to track performance and identify bottlenecks.

Table 2 illustrates that as the priority arrival rate (ϖ) mount, $L_q$ and ϒ also elevates, however ${\mathrm{\Pi }}_0$, ${\mathcal{P}}_0$ declines for the value of $\zeta =2.92$, ${\mathcal{I}}_1$=0.010, ${\mathcal{I}}_2$=0.117, b=0.99, γ=2, s=0.05, p= 0.98, ${\mathcal{D}}^{⁎}(\zeta +\varpi )$=0.5, ${\mu }_b^{(1)}$ = 1.04 and ${\mu }_b^{(2)}$ = 1.087. Moreover, a graphical depiction of the priority arrival rate mounts for increasing value of $L_q$ and ${\mathrm{\Gamma }}_1$ while decreasing ${\mathrm{\Pi }}_0$ which is presented in Fig. 4(b). Similarly, with increasing priority arrivals (ϖ), the software must allocate more resources to handle the larger queue $(L_q)$ while ensuring prompt service for priority patients. Optimized algorithms reduce wait times by prioritizing high-priority cases, addressing the higher retrial rate (ϒ) and ensuring efficient handling of increased demand. Decreasing server inactivity during normal $({\mathcal{P}}_0)$ and slow service $({\mathrm{\Pi }}_0)$ period underscores higher system utilization, especially for priority patients, emphasizing the need for efficient scheduling algorithms.

Table 3 displays that as the retrial rate (φ) elevates, ${\mathrm{\Pi }}_0$, $L_q$, ${\mathrm{\Gamma }}_2$ and ${\mathrm{\Gamma }}_b$ declines for the value of $\zeta =1$, ${\mathcal{I}}_1$=1, ${\mathcal{I}}_2$=1, b=0.8, ϖ=1, γ=6, s=0.5, p= 0.4, ${\mu }_b^{(1)}$ = 0.25 and ${\mu }_b^{(2)}$ = 0.0625. Here, Fig. 4(c) depicts ${\mathrm{\Pi }}_0$, $L_q$ and ${\mathrm{\Gamma }}_2$ and ${\mathrm{\Gamma }}_b$ falls down as the retrial rate rises. As the retrial rate increases, more patients attempt to retry for service, leading to faster service. This adaptation leads to quicker service and shorter queue lengths, enhancing overall efficiency. Moreover, the software optimizes resource utilization, ensuring fewer periods of server inactivity even amidst heightened demand. Thus, the observed improvements in queue length, service rate, and server activity directly result from the software's adept management, ultimately enhancing the hospital's operational effectiveness.

Table 4 illustrates that as the WV rate $({\chi }_v)$ elevates, ϒ also rises, however $L_s$, $L_q$ and ${\mathrm{\Pi }}_0$ declines for the value of $\zeta =2.1$, ${\mathcal{I}}_1$=0.47, ${\mathcal{I}}_1$=0.22, b=0.008, ϖ=8, s=0.01, p= 0.768, ${\mu }_p^{(1)}$ = 0.60, ${\mu }_p^{(2)}$ = 0.36, ${\mu }_b^{(1)}$ = 0.79 and ${\mu }_b^{(2)}$ = 0.629. Further, ϒ rises and $L_s$, $L_q$ and ${\mathrm{\Pi }}_0$ declines as the lower service rate rises which is presented in Fig. 4(d). As the lower service rate $({\chi }_v)$ increases, patients encounter busy servers, prompting them to retry for service, thereby increasing the retrial rate (ϒ). In order to mitigate this, HMS software optimises patient flow, minimises patient annoyance, and implements effective retry management techniques. Patients are being treated more quickly, as seen by the decline in system length $(L_q)$ and queue length $(L_s)$, which is a result of the software's capacity to optimise allocation of funds and optimise workflows. Furthermore, the decline in the probability of server idleness $({\mathrm{\Pi }}_0)$ emphasises how the programme contributes to improved staff efficiency and consumption of resources. Thus, HMS raises the standard of care given by guaranteeing continual server participation.

Table 5 depicts that as the balking probability (b) sums up, ${\mathrm{\Pi }}_0$, ϒ and ${\mathrm{\Gamma }}_1$ also rises, however $W_q$ diminish for the value of $\zeta =2.1$, ${\mathcal{I}}_1$=0.47, ${\mathcal{I}}_1$=0.22, γ=2.5, ϖ=8, s=0.01, p= 0.768, ${\mu }_p^{(1)}$ = 0.60, ${\mu }_p^{(2)}$ = 0.36, ${\mu }_b^{(1)}$ = 0.79 and ${\mu }_b^{(2)}$ = 0.629. As the balking probability (b) increases, more patients opt to leave the queue without service, leading to a higher retrial rate (ϒ) as patients attempt to retry. HMS software effectively manages this by optimizing patient flow and minimizing wait times. The increase in service rate $({\mathrm{\Gamma }}_1)$ indicates quicker service, likely due to improved resource allocation facilitated by the software. Consequently, there's an increase in the probability of server inactivity $({\mathrm{\Pi }}_0)$, signalling opportunities for further resource optimization. Faster service also translates to reduced waiting time $(W_q)$, enhancing patient satisfaction and overall efficiency, all managed effectively by the software.

Table 6 displays that as the reneging probability (s) rises up, ${\mathrm{\Pi }}_0$ and ${\mathrm{\Gamma }}_b$ rises but $W_q$ and $W_s$ declines, for the value of $\zeta =2.1$, ${\mathcal{I}}_1$=0.47, ${\mathcal{I}}_1$=0.22, b=0.008, ϖ=8, b=0.8, p= 0.768, ${\mu }_p^{(1)}$ = 0.60, ${\mu }_p^{(2)}$ = 0.36, ${\mu }_b^{(1)}$ = 0.79 and ${\mu }_b^{(2)}$ = 0.629. As the reneging probability increases, more patients leave the queue after some time without receiving service, reducing queue $(W_q)$ and system $(W_s)$ congestion and decreasing waiting times. This indicates that hospital management software can effectively manage patient flow and reduce wait times. The increase in service rate $({\mathrm{\Gamma }}_b)$ suggests that patients are being served more quickly, likely due to improved resource allocation and streamlined processes facilitated by the software. Additionally, the increase in the probability of server inactivity $({\mathrm{\Pi }}_0)$ due to the high service rate highlights opportunities for further optimization within the healthcare facility. The software can identify and address periods of resource underutilization, ensuring more consistent and efficient use of resources.

In addition, Fig. 5(a) - 5(f) addresses a few 3D graphs. Fig. 5(a) and Fig. 5(d) shows that as arrival rate (ϖ) mounts, $L_q$, ${\mathrm{\Gamma }}_1$ and ϒ also elevates, whereas ${\mathcal{P}}_0$ falls. With more patients arriving, the queue length increases, requiring hospital management software to optimize patient flow. The higher arrival rate leads to a busy server and increased retrial rates, necessitating efficient retry management. Additionally, to accommodate increased demand, the service process needs to be expedited, resulting in a higher rate of patients being served. This requires the software to dynamically allocate resources and adjust schedules to maintain high service rates. Despite longer queues, the probability of server inactivity decreases, indicating consistent resource utilization. Continuous monitoring and adjustment by the software ensure efficient and productive hospital operations.

Figure 5.

3D representation of effects of various parameters.

Fig. 5(b) and Fig. 5(e) depicts that as priority arrival rate (ζ) rises, $L_q$ and ϒ also mounts, however ${\mathcal{P}}_0$, ${\mathrm{\Pi }}_0$ and declines. The increase in Lq for priority patients highlights the need for efficient handling and prioritization in the queue to manage the higher influx of critical cases. The higher retrial rate underscores the necessity for HMS to implement robust mechanisms to handle retries efficiently, ensuring priority patients receive timely attention. This decrease in server inactivity is more pronounced for priority arrivals, emphasizing the importance of having dedicated resources or processes to handle priority patients effectively to maintain continuous server activity.

As retrial rate (φ) rises in Fig. 5(c), $L_q$ and ${\mathrm{\Pi }}_0$ declines. A decrease in queue length indicates more efficient patient inflow and faster service rates. HMS should use real-time data analytics to optimize patient flow and maintain manageable queue lengths, even during peak times. With reduced server inactivity, hospital efficiency and productivity improve. The software should offer comprehensive reporting and analytics to help administrators identify and implement continuous process improvements.

Finally, in Fig. 5(f) as lower service rate ${\chi }_v$ mounts, ϒ also elevates whereas $L_s$ falls. Despite more time-consuming cases being handled and an increase in patient retries, the system length declines, indicating efficient patient processing. Continuous monitoring and optimization of patient flow and resource utilization by the software are essential. The rising retrial rate with declining system length suggests effective management of retries without increasing patient wait times.

5.1. ANFIS computing

Adaptive Neuro-Fuzzy Inference Systems (ANFIS) combine fuzzy logic and neural networks to create a robust tool for modeling and control applications developed by Jang [10] the early 1990s. ANFIS utilizes fuzzy sets and membership functions to handle uncertainty and linguistic variables, while employing fuzzy rules to describe input-output relationships. Its structure consists of five layers: input nodes for membership functions, rule nodes for computing rule firing strengths, normalization nodes for normalizing these strengths, consequence nodes for generating weighted consequent values, and an output node for aggregating these values. The system uses a hybrid learning algorithm, combining least squares estimation and gradient descent, to optimize parameters, enhancing its capability to model complex, non-linear systems accurately. This integration enables ANFIS to adapt to dynamic environments and learn from data, making it suitable for applications such as control systems, pattern recognition, data prediction, and more. Its effectiveness in providing precise, adaptable solutions to intricate problems underscores its wide-ranging utility in various fields.

This approach has been adopted recently by Madhu Jain and Sibasish Dhibar [11] and Vaishnawi et al. [35]. All things considered, ANFIS provides the ability to handle the data in a dynamic way. Below the general ANFIS structure with two input variables is illustrated in Fig. 6.

Figure 6.

ANFIS structure.

We have juxtaposed the neuro-fuzzy results of the SVT/PGF analysis with those of the ANFIS technology. Connecting a fuzzy approach to ANFIS networks requires assessing a few aspects as inputs, which are observed as linguistic terms. By adjusting the arrival rate ϖ, priority arrival rate ζ, and lower service rate ${\chi }_v$ with fixed levels of $L_S$, we have demonstrated three distinctive variations in mean system size here. 3 linguistic values are added to the parameters, which are thought of as linguistic variables accumulated over 3 epochs, in order to obtain the intended result utilizing the ANFIS technique. All these linguistic variables ϖ, ζ, and ${\chi }_v$ are presumed to have Gaussian membership functions as their membership functions, which is illustrated in Fig. 7 in their respective forms. The three membership functions that are employed here are low, medium, and high which define the linguistic terms' degrees. The membership function count and the linguistic values of the relevant input parameters are listed in Table 7.

Figure 7.

Membership functions.

Table 7.

Membership function values depending on the input parameter.

Input parameters	No. of membership functions	Linguistic Values
ϖ, Ls	3	Low, Medium, High
ζ, Ls	3	Low, Medium, High
χv, Ls	3	Low, Medium, High

The MATLAB application is employed to obtain analytical results for ANFIS. By fixing $L_s$, the range of $L_s$ from 2 to 13.5 has been approximated in relation to the values of ϖ, ζ, and ${\chi }_v$. In Fig. 8, the blue line symbolises the ANFIS result, while the red line symbolises the exponential function. Fig. 9 illustrates how ANFIS has been used to represent the ϖ, ζ, and ${\chi }_v$ in 3D. In the end, we discovered that the outcomes generated by the ANFIS and exponential functions were akin.

Figure 8.

Modifications in L_s for the distinct values of (a) ϖ, (b) ζ and (c) χ_v.

Figure 9.

ANFIS based 3D visualization.

6. Cost analysis

Cost analysis is the practice of assessing the expenses related to a specific project or operation. It entails classifying costs into groups and evaluating them to see how they affect the budget as a whole. We employ the cost optimization strategy to obtain the ideal parameters, i.e., service rates (${\tau }_b$, ${\tau }_v$). Thus, we presumed that the cost structure of the predicted cost function is linear in terms of the cost components linked to the varied system operations.

The preceding definitions apply to cost factors that are part of the predicted total cost function TC (${\tau }_b$, ${\tau }_v$) for per unit time (PUT):

${\mathcal{G}}_h$	-	Each patient's holding cost PUT utilised in the system
${\mathcal{G}}_b$	-	Cost PUT during the server's active mode
${\mathcal{G}}_v$	-	Cost PUT during the server's WV phase
${\mathcal{G}}_1$	-	Cost per priority patient treated in busy hours
${\mathcal{G}}_2$	-	Cost per priority patient treated in server's WV mode

As a result, the notion of the predicted cost function is states as below,

$$TC={\mathcal{G}}_h{L}_q+{\mathcal{G}}_b({\mathrm{\Gamma }}_1+{\mathrm{\Gamma }}_2+{\mathrm{\Gamma }}_b)+{\mathcal{G}}_v{\mathrm{\Lambda }}_v+{\mathcal{G}}_1{\tau }_b+{\mathcal{G}}_2{\tau }_v$$ (6.45)

The cost function indicated by (6.45) is substantially non-linear, thus rendering its analytical optimization tricky. Thereby, we implement heuristic techniques to optimize overall cost, which we deduce to be a function of service rates (${\tau }_b$, ${\tau }_v$).

6.1. Cost optimization

Cost optimisation is the strategic process of assessing and improving an organization's financial performance through prudent expense management. Achieving the lowest value feasible will guarantee that resources are distributed wisely in support of the organization's long-term objectives. A company can increase both its competitive advantage and financial stability by incorporating cost optimisation into its operations. Here, the primary goal is to minimise the TC function in order to estimate the optimal service rates, ${\tau }_b^{⁎}$ and ${\tau }_v^{⁎}$, for the server's active state and WV mode, respectively.

In terms of math, the cost-minimization problem is expressed as follows,

$$TC({\tau }_b^{⁎},{\tau }_v^{⁎})=\underset{{\tau }_b^{⁎},{\tau }_v^{⁎}}{Minimize}TC({\tau }_b,{\tau }_v)$$

Table 8 lists the ranges of the cost elements.

Table 8.

Different cost sets for cost analysis.

Cost sets	${\mathcal{G}}_h$	${\mathcal{G}}_b$	${\mathcal{G}}_v$	${\mathcal{G}}_1$	${\mathcal{G}}_2$	${\mathcal{G}}_3$	${\mathcal{G}}_4$
Set 1	15	95	45	55	30	30	20
Set 2	20	105	35	25	30	20	35
Set 3	15	120	35	25	22	10	15
Set 4	20	100	30	45	15	20	17

6.2. Particle swarm optimization (PSO)

Particle Swarm Optimization (PSO) was introduced by Kennedy and Eberhart [14] is an optimization technique inspired by the behaviour of birds or fish. It involves a group of candidate solutions wandering around a solution space. Further, every one of them adjusts their positions and velocities in accordance with its personal observation and the ideal spots discovered by others, thereby iteratively converging on an optimal or near-optimal solution. This strategy was further developed by Upadhyaya [32] to address cost optimization in a discrete-time RQ with Bernoulli feedback and initial failure. Zhang et al. [38] developed computational and cost solutions for a single-server recurrent system with state-dependent service by employing a PSO algorithmic program. The research carried out by Malik et al. [19] has been quoted for deeper information on PSO functioning.

Here, ζ=1.2, ϖ=1, ${\mathcal{D}}^{⁎}(\zeta +\varpi )$=0.5, s =0.5 are few system parameters whose values are set in order to optimise the overall cost. Additionally, ${\tau }_b$ has an upper bound of 4 and ${\tau }_v$ has lower bound of 1. Furthermore, there are 100 iterations, two acceleration factors namely 1 and 2 are assumed, and 50 and 1 are the values for the population size and inertial weight, respectively. Additionally, The pseudocoded sequence of operations for the PSO algorithm is laid forth in Algorithm 1. With its aid and the cost values mentioned in Table 8 we derived the Table 9 which clearly details the effect of $b,\phi ,{\chi }_b$ on TC via PSO, on the ideal service rates $({\tau }_b⁎,{\tau }_v⁎)$.

Algorithm 1.

PSO Algorithm Pseudo Code.

Table 9.

Effect of b,φ,χ_b on (TC in $) using PSO.

Parameters			(TC in $)
		Cost set 1  Cost set 2  Cost set 3 Cost set 4
	0.85	(151.5876)	(159.3748)	(129.4700)	(179.2653)
b	0.86	(148.9678)	(157.0311)	(127.5690)	(176.0622)
	0.87	(146.2267)	(154.5817)	(125.5809)	(172.7066)
	12.5	(168.9862)	(123.3129)	(151.7573)	(143.1213)
φ	12.6	(171.1178)	(124.5895)	(153.3365)	(144.8767)
	12.7	(173.2104)	(125.8428)	(154.8871)	(146.6002)
	1.81	(176.8373)	(128.0217)	(157.5857)	(149.5939)
χb	1.82	(174.3785)	(126.5564)	(155.7761)	(147.5762)
	1.83	(171.8887)	(125.0739)	(153.9458)	(145.5343)

6.3. Artificial Bee Colony optimization (ABC)

Artificial Bee Colony (ABC) is an optimization algorithm proposed by Karaboga and Basturk [13] in light of honeybee foraging behaviour. It makes use of employee bees to investigate potential solutions, onlooker bees to identify the best ones, and scout bees to discover fresh ones. This iterative method effectively finds optimal or near-optimal solutions to complex optimization problems.

Here, the settings of a few system parameters—ζ=1.2, ϖ=1, ${\mathcal{L}}_v^{⁎}(\gamma )$=2, and b = 0.8 —are chosen to optimise the total cost. Furthermore, the lower bound for ${\tau }_v$ is 2, whereas the upper bound for ${\tau }_b$ is 5. In addition, there are 50 iterations with acceleration co-efficient 1 and the population size 50 are presumed. The pseudocoded sequence of operations for the ABC method is also presented in Algorithm 2. Thus, Table 10 is been constructed, which shows in clear detail the impact of $b,\phi ,{\chi }_b$ on total minimal cost TC utilising ABC optimisation methodologies, on the optimal service rate pairings $({\tau }_b⁎,{\tau }_v⁎)$.

Algorithm 2.

ABC Algorithm Pseudo Code.

Table 10.

Effect of b,φ,χ_b on (TC in $) using ABC.

Parameters			(TC in $)
		Cost set 1  Cost set 2  Cost set 3 Cost set 4
	0.85	(129.7364)	(153.6764)	(171.3470)	(146.7485)
b	0.86	(134.1254)	(159.4993)	(176.8652)	(151.0893)
	0.87	(138.5234)	(165.3160)	(182.4067)	(155.4408)
	12.5	(164.4035)	(193.8496)	(131.64603)	(147.5766)
φ	12.6	(166.0766)	(196.0303)	(131.6460)	(149.2610)
	12.7	(167.7445)	(198.2051)	(135.1006)	(150.9400)
	1.81	(134.2011)	(154.8546)	(138.2505)	(133.3733)
χb	1.82	(134.8129)	(155.6592)	(139.0233)	(134.0549)
	1.83	(135.4306)	(156.4714)	(139.8036)	(134.7430)

6.4. Grey Wolf Optimiser (GWO)

Grey Wolf Optimisation (GWO) was developed by Mirjalili et al. [20] in response to the social structure and hunting techniques of grey wolves. Alpha, beta, delta, and omega wolves are employed to steer and broaden the search. GWO efficiently discovers optimal or almost optimal solutions to challenging issues by imitating the hunting strategies of encircling, pursuing, and attacking prey.

Here, the system parameters namely ζ=1.2, ϖ=1, ${\mathcal{D}}^{⁎}(\zeta +\varpi )$=0.5, and s = 0.5 are utilized to optimize the cost of the proposed framework. Further, ${\tau }_b$ has an upper bound of 10 and a lower bound of 1 for ${\tau }_v$. In addition, the total no. of iterations is presumed to be 100 along with the population size 100. In addition, the pseudocoded sequence of operations for the GWO method is presented in Algorithm 3. Therefore, we generated Table 11 with the assumed values and the cost values listed in Table 8. It shows in explicit detail the impact of $b,\phi ,{\chi }_b$ on the total minimal cost TC employing GWO optimisation techniques, on the optimal service rate pairs $({\tau }_b⁎,{\tau }_v⁎)$.

Algorithm 3.

GWO Algorithm Pseudo Code.

Table 11.

Effect of b,φ,χ_b on (TC in $) using GWO.

Parameters			(TC in $)
		Cost set 1  Cost set 2  Cost set 3 Cost set 4
	0.85	(152.7202)	(129.0756)	(116.2024)	(128.6166)
b	0.86	(153.5984)	(130.2436)	(117.0841)	(129.4956)
	0.87	(154.4838)	(131.4211)	(117.9731)	(130.3817)
	12.5	(160.8392)	(139.8504)	(124.3799)	(136.7474)
φ	12.6	(161.2378)	(140.3825)	(124.7777)	(137.1458)
	12.7	(161.6367)	(140.9151)	(125.1757)	(137.5445)
	1.81	(160.6383)	(139.6176)	(124.2539)	(136.5978)
χb	1.82	(160.4796)	(139.3472)	(124.0477)	(136.3932)
	1.83	(160.2789)	(139.0813)	(123.8449)	(136.1921)

6.5. Differential Evolution (DE)

Storn and Price [31] devised the optimisation algorithm known as Differential Evolution (DE). It involves a population of potential fixes that change with each cycle. By merging each candidate solution with others through some processes, each candidate solution is updated. DE is a popular option in many scientific and engineering applications because of its efficiency, simplicity, and capacity to manage challenging, non-linear optimisation issues.

Certainly, the system's optimization involves setting parameters such as ζ, ϖ, p, s and ${\mathcal{L}}_v^{⁎}(\gamma )$ to values like 1.2, 1, 0.4, 0.5 and 2 respectively, aimed at minimizing overall cost. Constraints are imposed on ${\tau }_b$ (with an upper limit of 4) and ${\tau }_v$ (with a lower limit of 1). The optimization process runs for 100 iterations, considering crossover probability to be 0.2 with scaling factors having upper bound of 0.8 and lower bound of 0.2, along with a population size of 50. Thus, the DE algorithm, outlined in Algorithm 4, guides the optimization process. By applying this algorithm and utilizing the cost values provided in Table 8, we construct Table 12. This table vividly illustrates how variations in parameters such as b, φ, and ${\chi }_b$ influence the total minimal cost using DE optimization. It also presents the corresponding optimal service rate pairs $({\tau }_b,{\tau }_v)$.

Algorithm 4.

DE Algorithm Pseudo Code.

Table 12.

Effect of b,φ,χ_b on (TC in $) using DE.

Parameters			(TC in $)
		Cost set 1  Cost set 2  Cost set 3 Cost set 4
	0.85	(149.6617)	(118.3828)	(135.7222)	(158.0185)
b	0.86	(150.4928)	(119.7611)	(137.3754)	(159.3961)
	0.87	(151.3305)	(121.1504)	(139.0417)	(160.7847)
	12.5	(157.6085)	(131.5150)	(151.0761)	(171.1401)
φ	12.6	(157.9907)	(132.1536)	(151.6630)	(171.7788)
	12.7	(158.3732)	(132.7814)	(152.2254)	(172.4180)
	1.81	(157.5480)	(130.2621)	(148.0439)	(170.9504)
χb	1.82	(157.3491)	(130.0944)	(147.9307)	(170.6417)
	1.83	(157.1534)	(129.9213)	(147.8137)	(170.3282)

6.6. Comparative analysis within PSO, ABC, GWO and DE

In order to estimate the lowest potential cost using their individual MATLAB programs, we look at four methods in this paper: PSO, ABC, GWO, and DE. We study four different cost sets, as Table 8 illustrates. The MATLAB programs for each algorithm are run sequentially, yielding Table 9, Table 10, Table 11, Table 12.

The outputs of the four programs were quite comparable to one another. As a result, the four techniques' best answers are close to one another. This indicates that the heuristics discussed above offer reliable, ideal solutions. The data presented in Table 9, Table 10, Table 11, Table 12 reveals that the software modelling costs range from $116 to $179, with GWO possessing the lowest ideal cost value at $116.2024. As it turns out, we can employ any strategy to ascertain the optimal cost; nonetheless, as we compare for the suggested framework, GWO is a very successful method for ascertaining the best feasible cost. It works flawlessly in global queries, is easy to configure, and is unaffected by scaling changes in design variables.

As indicated in Table 9, Table 10, Table 11, Table 12, it is evident that the GWO method yields the lowest ideal cost value. This observation is further supported by the findings in Fig. 10, which demonstrate the influence of system parameters on Total Cost (TC) across the four distinct techniques via 2D graphs.

Figure 10.

Influence of system parameters on TC using distinct cost sets is depicted via two dimensional graphs.

Figs. 10(a) and 10(b) illustrate the impact of balking probability (b) on TC using cost sets 3 and 4, respectively. It is evident from both figures that GWO consistently achieves the lowest cost value. Additionally, Figs. 10(c), 10(d), and 10(e) showcase the effect of the retrial rate on TC using cost sets 1, 2, and 3, respectively. While Figs. 10(c) and 10(d) indicate that DE and PSO yield the lowest ideal cost, Fig. 10(e) highlights GWO as having the lowest cost value once again. Figs. 10(f) and 10(g) depict the impact of the normal service rate on TC using cost sets 3 and 4, respectively. Fig. 10(f) clearly demonstrates that GWO consistently yields the lowest cost value. However, a slight deviation is observed in Fig. 10(g) where there is a minor discrepancy between the lowest cost values of ABC and GWO. Nonetheless, GWO ultimately emerges with the lowest ideal cost.

Thus, based on the findings in Fig. 10, it can be concluded that GWO not only provides the lowest possible cost but also exhibits faster convergence compared to the other three methods.

6.7. Convergence in PSO, ABC, GWO and DE

In metaheuristic algorithms, convergence is crucial because it means that the algorithm is getting close to finding an ideal or nearly ideal solution. Because it facilitates the attainment of dependable and precise outcomes in a reasonable amount of time, the method is beneficial and practical for real-world applications. Thus, the typical results of applying the techniques PSO, ABC, GWO, and DE to each of the cost sets specified in Table 8 are shown in Table 9, Table 10, Table 11, Table 12. Additionally, as Fig. 11 illustrates, after a given number of trials (generations) in PSO, ABC, GWO, and DE, particles converge to the best solution. The particles in all four approaches converge on the lowest cost after a short number of instance generations. As a whole, it appears from the Fig. 11 that GWO converges more quickly. We recognize that the criteria selected for vacation, voluntary service, and priority have an impact on the overall anticipated cost, and administrators should proceed with extreme caution when making these decisions. The analysts' financial burden will be partially mitigated by their ability to purchase this software in its whole.

Figure 11.

Convergence of the cost function.

7. Conclusion

The identified research gaps and the primary managerial implications related to HMS software is stated below:

• •
  Main Gaps and Limitations:

  • –
    The existing literature lacked studies focusing on RQ frameworks specifically tailored for HMS.
  • –
    Previous research inadequately explored batch arrival RQ under BWV conditions. This omission left a significant gap in understanding how batch arrivals and retrials impact HMS performance and efficiency.
  • –
    Prior studies did not comprehensively analyze the influence of various system parameters on TC in HMS. This limitation hindered the ability to make informed decisions regarding system optimization and resource allocation.
• •
  Managerial Implications:

  • –
    Implementing an effective HMS software can streamline administrative tasks such as patient registration, appointment scheduling, and billing processes.
  • –
    Managers can utilize the software to automate routine tasks, reducing manual errors and administrative burdens on staff.
  • –
    By applying optimization techniques one can identify optimal solutions that minimize costs while maximizing productivity and service quality which leads to significant savings in operational expenses, improved financial performance, and enhanced overall efficiency within the healthcare organization.

From the aforementioned lines, we can say that by modelling a priority retrial queuing framework with impatient customers offers valuable insights into HMS software, shedding light on patient flow, service provision, and resource utilization complexities. Further, by incorporating SVT underscores its importance in modeling HMS software, as evidenced by its impact on system performance and efficiency in numerical examples. Also, inclusion of BWV enables more accurate healthcare system representations, aiding resource allocation and patient management decisions. The improved performance observed underscores the potential benefits of advanced queuing frameworks in real-world healthcare settings. Moreover, by comparing neuro-fuzzy results with analytical findings using ANFIS, we ensure model accuracy and reliability, enhancing its credibility for real-world application. Finally, the optimization techniques used provide robust computational tools for addressing complex problems, leading to better patient outcomes and enhanced quality of care in HMS operations.

Therefore, the study addresses critical gaps in HMS research alongside queueing framework, providing valuable insights and practical implications for healthcare managers striving to optimize system efficiency and cost-effectiveness.

Future research could expand on this study by modelling distinct real life systems such as production facility, software computing system as distinct QS. Moreover, the proposed QS can also be studied by considering batch service of patients. Further, transient analysis of the retrial queueing framework was potentially a viable field for further research.

Funding

We would like to express our sincere gratitude to Vellore Institute of Technology, Vellore, India for their generous financial support, which made this research possible. Thanking again for their unwavering support and encouragement.

CRediT authorship contribution statement

R. Harini: Writing – original draft. K. Indhira: Writing – review & editing, Validation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

No data was used for the research described in the article.