Analysis of a GI/M/1/N Retrial Queue with Working Vacations, Bernoulli Feedback, and Impatient Customers: Steady-State Analysis, Asymptotics, and Cost Optimization for Next-Generation Networks

1. Introduction

Queueing theory has emerged as a powerful mathematical framework for analyzing and mitigating congestion in telecommunications networks, healthcare facilities, banking services, call centers, and manufacturing systems [1,2,3,4]. Understanding and optimizing the performance of such systems under various operational conditions is crucial for enhancing service quality, reducing waiting times, and minimizing operational costs.

The concept of a working vacation, introduced by Servi and Finn [5], allows the server to continue serving customers at a reduced rate during idle periods rather than stopping service completely. This concept has subsequently been extended to more complex scenarios, including GI/M/1 queues [6], finite buffer systems [7], and queues with customer impatience [8,9,10,11]. Prolonged delays can also induce balking (an arriving customer chooses not to join) and reneging (a waiting customer leaves before service). Such impatience can significantly degrade performance and generate substantial losses. To mitigate this, organizations use retention strategies—special offers or improved service guarantees—to persuade impatient customers to stay [12,13,14]. The retention mechanism, whereby a reneging customer is persuaded to stay with probability $\overline{\alpha }$, has been incorporated into several queueing models [15]. Customer feedback is another common phenomenon: after service completion, a customer either leaves the system permanently with probability $\beta$ or rejoins the system for another service round with probability $\overline{\beta }$ [13,16]. Feedback has been studied extensively in retrial [17,18] and vacation [19] contexts.

In contrast to classical models in which customers wait in a single physical queue, retrial systems are characterized by the fact that customers who find the server busy join a virtual pool, called the orbit, and re-attempt service after a random time [20,21]. This behavior is typical of telephone systems, call centers, and random-access protocols. Under the classical linear retrial policy, each orbiting customer retries independently after an exponential time with rate $\eta$, so that the total retrial attempt rate is $r\eta$ when r customers are in the orbit. The alternative constant retrial policy, in which the orbit-wide attempt rate is a constant independent of r, is also a legitimate modeling assumption in the literature [21]; the choice between the two is a modeling decision, and this paper adopts the linear policy, which is the physically natural one for independent, unsynchronized customers.

Recently, Guendouzi and Saadaoui [19] analyzed a GI/M(k)/1/N queueing system with multiple working vacations, Bernoulli feedback, and customer impatience with retention, in which customers wait in a physical queue. In many practical situations, particularly in telecommunication systems and call centers, customers who encounter a busy server instead join a retrial orbit. The combination of retrial behavior with working vacation policies has attracted increasing attention [17,18]; yet the combination of a retrial orbit with working vacations, Bernoulli feedback, state-dependent service rates, and impatience with retention remains largely unexplored. Formulating such a model consistently requires care: the server may be idle while the orbit is non-empty, a possibility that must be reflected in the state space, and every transition that feeds or drains the orbit must appear in the governing equations. Providing such a formulation is one purpose of this paper.

Metaheuristic algorithms have proven highly effective for queueing optimization problems [22], including particle swarm optimization (PSO) [23], genetic algorithms (GA) [24], and simulated annealing (SA) [25]. These methods handle the non-linear, multi-modal objective functions that arise in queueing systems. In the context of 6G networks and cloud computing, “customer retention” relates directly to packet retransmission mechanisms, and the retrial orbit matches virtual waiting buffers in software-defined networks; in smart healthcare, appointment scheduling with patient impatience and automated call-back systems reflects the same dynamics [3,4]. Our analysis thus provides theoretical foundations for resource allocation in next-generation digital infrastructures.

This paper studies a single queueing model that integrates: (i) a general independent (GI) arrival process; (ii) a finite total capacity N; (iii) multiple working vacations; (iv) Bernoulli feedback; (v) customer impatience with retention; and (vi) a retrial orbit with linear (orbit-size-dependent) retrial rates. The main contributions are:

• a mathematically consistent formulation of the GI/M/1/N retrial queue with working vacations, feedback, and retention, with an explicit four-mode server state space in which the total retrial rate is $r\eta$;
• the steady-state distributions at pre-arrival and arbitrary epochs, obtained by the supplementary variable technique combined with the embedded arrival-epoch Markov chain, including closed-form expressions of the required matrix exponentials for deterministic, Erlang, and hyperexponential inter-arrival times;
• a degenerate-limit result showing that, as $\eta \to \infty$, the model reduces in distribution to the GI/M/1/N waiting-line model of [19], together with a numerical validation of Algorithm 1 against an independent discrete-event simulation;
• consistent performance measures, including an internal flow-balance identity that serves as a numerical correctness check;
• a stability discussion for the infinite-capacity analogue;
• an economic model without cost double counting, optimized by PSO, with comparative GA/SA experiments and a robustness analysis of the cost parameters;
• numerical illustrations and managerial insights, together with an economic comparison of the linear and constant retrial policies.

The remainder of the paper is organized as follows. Section 2 describes the model. Section 3 presents the governing equations, the steady-state solution, the computation algorithm, the validation limit, and stability remarks. Section 4 derives the performance measures. Section 5 develops the economic model and the optimization. Section 6 reports numerical results, and Section 7 concludes.

2. Model Description

We consider a single-server retrial queueing system with a finite total capacity N, multiple working vacations, Bernoulli feedback, and customer impatience with retention. There is no physical waiting line: at most one customer can be in service, and all waiting customers reside in the retrial orbit.

2.1. Assumptions

• Arrival process. Inter-arrival times are i.i.d. with CDF $A\left(x\right)$, density $a\left(x\right)$, LST $A^{*}\left(s\right)$, and mean $1/\lambda =-A^{*\prime }\left(0\right)$. Deterministic (D), Erlang ($E_k$), and hyperexponential ($H_2$) inter-arrival times are used in the numerical study.
• Capacity. The total number of customers in the system (in service plus in the orbit) never exceeds N. An arrival finding the system full is blocked (lost).
• Service. Service times are exponential. During working vacations the rate is ${\nu }_n$, and during normal busy periods it is ${\mu }_n$, where n is the total number of customers present (in service plus orbit) at that instant, $1\le n\le N$, with ${\mu }_n>{\nu }_n>0$. By memorylessness of the exponential distribution, the rate may change during a service when the orbit size changes.
• Working vacation policy. Whenever the system becomes completely empty (no customer in service and empty orbit), the server starts a working vacation of exponential duration with rate $\varphi$. During a vacation, arriving and retrying customers are served at rate ${\nu }_n$. When the vacation ends: if at least one customer is present (in service or in the orbit), the server switches to the normal mode (an ongoing exponential service simply continues at rate ${\mu }_n$); if the system is empty, a new vacation begins (multiple working vacations). While the server is idle in the normal mode with a non-empty orbit, it remains on duty at normal rates and no vacation clock runs.
• Retrial orbit (linear retrial policy). Each orbiting customer retries independently after an exponential time with rate $\eta$; hence the total attempt rate is $r\eta$ when r customers orbit. A retrial attempt that finds the server idle enters service immediately; otherwise the customer returns to the orbit (the state is unchanged). An arriving customer that finds the server busy with total occupancy $n=r+1\le N-1$ joins the orbit with probability ${\theta }_n$ or balks with probability ${\overline{\theta }}_n=1-{\theta }_n$ (${\theta }_N=0$); an arrival that finds the server idle with $r\le N-1$ starts service immediately.
• Impatience with retention. Each orbiting customer carries an independent exponential patience clock with rate $\xi$ (re-armed after each retention event). When a clock rings, the customer abandons the system with probability $\alpha$ or is retained in the orbit with probability $\overline{\alpha }=1-\alpha$.
• Bernoulli feedback. Upon a service completion, the customer leaves the system permanently with probability $\beta$ or rejoins the orbit with probability $\overline{\beta }=1-\beta$. Since the customer vacates the service position, feedback leaves the total occupancy unchanged and never violates the capacity constraint.
• Independence. All random variables involved are mutually independent.

2.2. System State

The state at time t is described by the Markov process $\left\{\right(S\left(t\right),R\left(t\right),U\left(t\right)),t\ge 0\}$, where $R\left(t\right)$ is the orbit size, $U\left(t\right)$ is the remaining inter-arrival time (the supplementary variable), and $S\left(t\right)\in \{0,1,2,3\}$ denotes the server mode:

$$S\left(t\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{idle},\mathrm{in}\mathrm{a}\mathrm{working}\mathrm{vacation},\hfill \\ 1,\hfill & \mathrm{busy},\mathrm{in}\mathrm{a}\mathrm{working}\mathrm{vacation},\hfill \\ 2,\hfill & \mathrm{busy},\mathrm{in}\mathrm{the}\mathrm{normal}\mathrm{mode},\hfill \\ 3,\hfill & \mathrm{idle},\mathrm{in}\mathrm{the}\mathrm{normal}\mathrm{mode}.\hfill \end{array}\right.$$

The feasible pairs $(S,r)$ are $(0,r)$, $0\le r\le N$; $(1,r)$ and $(2,r)$, $0\le r\le N-1$; and $(3,r)$, $1\le r\le N$, giving $4N+1$ macro-states. Note that the idle states $(0,r)$ and $(3,r)$ with $r\ge 1$ (idle server with a non-empty orbit) are precisely the hallmark of retrial systems: the server waits for the next retrial attempt rather than serving a physical queue. The instant at which the orbit empties while the server is idle in the normal mode triggers an immediate working vacation, so the pair $(3,0)$ is not a steady state.

2.3. Transition Summary

Table 1 lists every transition of the model; the governing equations of Section 3 are read off directly from it.

Table 2. Notation.

Symbol	Description
$\lambda$	mean arrival rate of the GI input
N	total capacity (service + orbit)
${\mu }_n,{\nu }_n$	service rates in normal / vacation mode, $n=$ total present
$\varphi$	working vacation rate (mean $1/\varphi$)
$\eta$	individual retrial rate per orbiting customer
$\xi$	impatience (patience clock) rate
$\alpha$ / $\overline{\alpha }$	abandonment / retention probability at a clock ring
$\beta$ / $\overline{\beta }$	final departure / feedback probability after service
${\theta }_n$ / ${\overline{\theta }}_n$	join-orbit / balk probability of a busy-server arrival, ${\theta }_N=0$

Table 2. Notation.

Symbol	Description
$\lambda$	mean arrival rate of the GI input
N	total capacity (service + orbit)
${\mu }_n,{\nu }_n$	service rates in normal / vacation mode, $n=$ total present
$\varphi$	working vacation rate (mean $1/\varphi$)
$\eta$	individual retrial rate per orbiting customer
$\xi$	impatience (patience clock) rate
$\alpha$ / $\overline{\alpha }$	abandonment / retention probability at a clock ring
$\beta$ / $\overline{\beta }$	final departure / feedback probability after service
${\theta }_n$ / ${\overline{\theta }}_n$	join-orbit / balk probability of a busy-server arrival, ${\theta }_N=0$

3. Steady-State Analysis

3.1. Governing Equations

For $j\in \{0,1,2,3\}$ and admissible r, let $P_{j,r}\left(u\right)du=Pr[S\left(t\right)=j,R\left(t\right)=r,u\le U\left(t\right)<u+du]$ in steady state, let $P_{j,r}^{\prime }\left(u\right)$ denote $d{P}_{j,r}\left(u\right)/du$, and define the arrival fluxes

$${\pi }_{j,r}:=P_{j,r}\left(0\right).$$

Between events, u decreases at unit speed; an arrival occurring in state $(j,r)$ resets u to a fresh inter-arrival time with density $a(·)$, which is why arrival terms appear as $a\left(u\right)\pi$. Observing the system at t and $t+dt$, taking limits, and using Table 1, we obtain the following differential-difference equations:

$$\begin{array}{cc}\hfill -P_{0,0}^{\prime }\left(u\right)& =\beta {\nu }_1{P}_{1,0}\left(u\right)+\beta {\mu }_1{P}_{2,0}\left(u\right)+\alpha \xi P_{0,1}\left(u\right)+\alpha \xi P_{3,1}\left(u\right),\hfill \\ \hfill -P_{0,r}^{\prime }\left(u\right)& =-\left[\varphi +r(\eta +\alpha \xi )\right]P_{0,r}\left(u\right)+\beta {\nu }_{r+1}{P}_{1,r}\left(u\right)+\overline{\beta }{\nu }_r{P}_{1,r-1}\left(u\right)+(r+1)\alpha \xi P_{0,r+1}\left(u\right),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & \hfill 1\le r\le N-1,\end{array}$$

(2)

$$\begin{array}{cc}\hfill -P_{0,N}^{\prime }\left(u\right)& =-\left[\varphi +N(\eta +\alpha \xi )\right]P_{0,N}\left(u\right)+\overline{\beta }{\nu }_N{P}_{1,N-1}\left(u\right)+a\left(u\right){\pi }_{0,N},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill -P_{1,0}^{\prime }\left(u\right)& =-({\nu }_1+\varphi )P_{1,0}\left(u\right)+\eta P_{0,1}\left(u\right)+\alpha \xi P_{1,1}\left(u\right)+a\left(u\right)\left[{\pi }_{0,0}+{\overline{\theta }}_1{\pi }_{1,0}\right],\hfill \\ \hfill -P_{1,r}^{\prime }\left(u\right)& =-\left[{\nu }_{r+1}+\varphi +r\alpha \xi \right]P_{1,r}\left(u\right)+(r+1)\eta P_{0,r+1}\left(u\right)+(r+1)\alpha \xi P_{1,r+1}\left(u\right)\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & +a\left(u\right)\left[{\pi }_{0,r}+{\theta }_r{\pi }_{1,r-1}+{\overline{\theta }}_{r+1}{\pi }_{1,r}\right],1\le r\le N-2,\hfill \\ \hfill -P_{1,N-1}^{\prime }\left(u\right)& =-\left[{\nu }_N+\varphi +(N-1)\alpha \xi \right]P_{1,N-1}\left(u\right)+N\eta P_{0,N}\left(u\right)\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & +a\left(u\right)\left[{\pi }_{0,N-1}+{\theta }_{N-1}{\pi }_{1,N-2}+{\pi }_{1,N-1}\right],\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill -P_{2,0}^{\prime }\left(u\right)& =-{\mu }_1{P}_{2,0}\left(u\right)+\varphi P_{1,0}\left(u\right)+\eta P_{3,1}\left(u\right)+\alpha \xi P_{2,1}\left(u\right)+a\left(u\right){\overline{\theta }}_1{\pi }_{2,0},\hfill \\ \hfill -P_{2,r}^{\prime }\left(u\right)& =-\left[{\mu }_{r+1}+r\alpha \xi \right]P_{2,r}\left(u\right)+\varphi P_{1,r}\left(u\right)+(r+1)\eta P_{3,r+1}\left(u\right)+(r+1)\alpha \xi P_{2,r+1}\left(u\right)\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & +a\left(u\right)\left[{\pi }_{3,r}+{\theta }_r{\pi }_{2,r-1}+{\overline{\theta }}_{r+1}{\pi }_{2,r}\right],1\le r\le N-2,\hfill \\ \hfill -P_{2,N-1}^{\prime }\left(u\right)& =-\left[{\mu }_N+(N-1)\alpha \xi \right]P_{2,N-1}\left(u\right)+\varphi P_{1,N-1}\left(u\right)+N\eta P_{3,N}\left(u\right)\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & +a\left(u\right)\left[{\pi }_{3,N-1}+{\theta }_{N-1}{\pi }_{2,N-2}+{\pi }_{2,N-1}\right],\hfill \\ \hfill -P_{3,r}^{\prime }\left(u\right)& =-r(\eta +\alpha \xi )P_{3,r}\left(u\right)+\varphi P_{0,r}\left(u\right)+\beta {\mu }_{r+1}{P}_{2,r}\left(u\right)+\overline{\beta }{\mu }_r{P}_{2,r-1}\left(u\right)+(r+1)\alpha \xi P_{3,r+1}\left(u\right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & \hfill 1\le r\le N-1,\end{array}$$

(10)

$$\begin{array}{cc}\hfill -P_{3,N}^{\prime }\left(u\right)& =-N(\eta +\alpha \xi )P_{3,N}\left(u\right)+\varphi P_{0,N}\left(u\right)+\overline{\beta }{\mu }_N{P}_{2,N-1}\left(u\right)+a\left(u\right){\pi }_{3,N}.\hfill \end{array}$$

(11)

Each negative term is a state-changing outflow (service, switch, retrial, reneging, arrival); each positive term is an inflow from Table 1. In particular, retrials from busy states return the customer to the orbit and hence do not appear as outflows of the busy states, and reneging from the orbit appears as $r\alpha \xi$ in every state class. The normalization is

$${\displaystyle \sum _{r=0}^N}{P}_{0,r}+{\displaystyle \sum _{r=0}^{N-1}}\left(P_{1,r}+P_{2,r}\right)+{\displaystyle \sum _{r=1}^N}{P}_{3,r}=1,\sum _{j,r}{\pi }_{j,r}=\lambda .$$

(12)

3.2. Pre-Arrival Epoch Probabilities

Lemma 1.

$P_{j,r}^{-}=P_{j,r}\left(0\right)/\lambda ={\pi }_{j,r}/\lambda$.

Proof. 

The probability flux through $u=0$ in state $(j,r)$ equals $P_{j,r}\left(0\right)$, since u decreases at unit speed. Each such flux event is an arrival that sees $(j,r)$. Hence the rate at which arrivals see $(j,r)$ is ${\pi }_{j,r}$, and dividing by the arrival rate $\lambda$ yields the probability that an arbitrary arrival sees $(j,r)$.    □

3.3. The Steady-State Solution

Between arrivals, $\left(S\right(t),R(t\left)\right)$ is a continuous-time Markov chain on the $4N+1$ macro-states with generator $\mathbf{Q}$ read off from the Markov rows of Table 1. Let $\mathbf{B}$ be the stochastic “action” matrix whose entry $\mathbf{B}[(j,r),(j^{\prime },r^{\prime })]$ is the probability that, immediately after an arrival that saw $(j,r)$, the macro-state is $(j^{\prime },r^{\prime })$ (the arrival rows of Table 1). Define

$$\mathbf{M}={\int }_0^{\infty }a\left(x\right)e^{\mathbf{Q}x}dx,\mathbf{G}={\int }_0^{\infty }\overline{A}\left(t\right)e^{\mathbf{Q}t}dt,\overline{A}\left(t\right)=Pr[X>t],$$

where $e^{\mathbf{Q}x}$ is the matrix exponential, and let $\mathbf{K}=\mathbf{B}\mathbf{M}$.

Theorem 1.

Assume $\mathbf{K}$ admits a unique stationary distribution (e.g., ${\theta }_n>0$ for $1\le n\le N-1$ and $\overline{\beta }>0$). Let $\mathbf{y}=\left(y_{j,r}\right)$ solve $\mathbf{y}=\mathbf{y}\mathbf{K}$, $\sum y_{j,r}=1$. Then, in steady state:

(i) 

$P_{j,r}^{-}=y_{j,r}$ (pre-arrival probabilities);

(ii) 

${\pi }_{j,r}=\lambda y_{j,r}$ (arrival fluxes);

(iii) 

$P_{j,r}=\lambda \left(\mathbf{y}\mathbf{B}\right)\mathbf{G}$ evaluated at the entry of $(j,r)$ (arbitrary-epoch probabilities), and the normalization (12) holds.

Proof (sketch).

The process regenerates at arrival epochs: given the pre-arrival state $Y_n$, the post-action state has distribution $\mathbf{y}\mathbf{B}$ row-wise, and the pre-arrival state $Y_{n+1}$ is obtained after the CTMC evolves through an independent inter-arrival time, giving the transition matrix $\mathbf{K}=\mathbf{B}\mathbf{M}$; stationarity gives (i) and (ii). For (iii), by the renewal–reward theorem, $P_{j,r}$ equals the expected total time spent in $(j,r)$ during one inter-arrival cycle divided by the mean cycle length $1/\lambda$; conditional on the post-action state s, the expected time in $(j,r)$ is ${\int }_0^{\infty }\overline{A}\left(t\right){\left[e^{\mathbf{Q}t}\right]}_{s,(j,r)}dt$, i.e., the entry of $\mathbf{G}$. Summing over s with weights ${\left(\mathbf{y}\mathbf{B}\right)}_s$ and multiplying by $\lambda$ gives (iii).    □

Proposition 1 (Closed forms for D, $E_k$, $H_2$).

Let ${\gamma }_i>0$ and weights $p_i$ ($\sum p_i=1$).

• D (inter-arrival time c): $\mathbf{M}=e^{\mathbf{Q}c}$, $\mathbf{G}={\int }_0^c{e}^{\mathbf{Q}t}dt$.
• $H_2$ ($a\left(x\right)={\sum }_i{p}_i{\gamma }_i{e}^{-{\gamma }_{i}x}$): $\mathbf{M}={\sum }_i{p}_i{\gamma }_i{({\gamma }_i\mathbf{I}-\mathbf{Q})}^{-1}$, $\mathbf{G}={\sum }_i{p}_i{({\gamma }_i\mathbf{I}-\mathbf{Q})}^{-1}$.
• $E_k\left(\gamma \right)$ (mean $k/\gamma$): $\mathbf{M}={\left[\gamma {(\gamma \mathbf{I}-\mathbf{Q})}^{-1}\right]}^k$, $\mathbf{G}={\sum }_{m=0}^{k-1}{\gamma }^m{(\gamma \mathbf{I}-\mathbf{Q})}^{-(m+1)}$.

All expressions follow from ${\int }_0^{\infty }{e}^{-\gamma x}{e}^{\mathbf{Q}x}dx={(\gamma \mathbf{I}-\mathbf{Q})}^{-1}$ and its derivatives; matrix exponential integrals may also be computed by Van Loan’s block-matrix technique [26].

3.4. Computation Algorithm

Algorithm 1 (steady-state computation).

1.

Construct $\mathbf{Q}$ and $\mathbf{B}$ (size $4N+1$) from Table 1.

2.

Compute $\mathbf{M}$ and $\mathbf{G}$ using Proposition 1 (or quadrature plus matrix exponentials for general a).

3.

Solve the sparse linear system $\mathbf{y}(\mathbf{I}-\mathbf{K})=0$, $\sum y_{j,r}=1$.

4.

Set $\pi =\lambda \mathbf{y}$ and $\mathbf{P}=\lambda \left(\mathbf{y}\mathbf{B}\right)\mathbf{G}$.

5.

Compute the performance measures of Section 4 and verify the flow identity (26) as a correctness check.

The cost is dominated by $O\left(N^3\right)$ matrix operations, which is negligible for $N\le 50$ (implementation in Python with NumPy/SciPy, provided as supplementary material).

3.5. Validation Limit: $\eta \to \infty$

Proposition 2.

As $\eta \to \infty$, the probabilities of the idle states with non-empty orbit vanish, $P_{0,r},P_{3,r}\to 0$ for $r\ge 1$, and the distribution of the total number of customers n converges to that of the GI/M/1/N waiting-line queue of [19] with the same ${\mu }_n,{\nu }_n,{\theta }_n,\beta ,\alpha ,\xi ,\varphi$.

Proof (sketch).

With instantaneous retrials the server is never idle while the orbit is non-empty, and a blocked/busy-server arrival effectively occupies a waiting position that seizes the server the moment it frees. Since all durations are exponential and service rates depend only on the total count, the order of service is immaterial for the count process, whose birth rates (arrivals accepted with ${\theta }_n$) and death rates (completions ${\nu }_n$/${\mu }_n$, feedback as a self-loop, reneging $(n-1)\alpha \xi$) coincide with those of the waiting-line model of [19]. □

Proposition 2 provides a theoretical validation anchor for future work with access to [19]’s implementation; Section 6 instead validates Algorithm 1 directly against an independent discrete-event simulation of the retrial system itself.

3.6. Stability and the Limit $N\to \infty$

For finite N the chain is finite, hence positive recurrent. For the infinite-capacity analogue: if $\alpha >0$, reneging drains the orbit at rate $r\alpha \xi$, which dominates any bounded inflow, so the system is stable for every $\lambda$; if $\alpha =0$, a sufficient condition is $\lambda {\overline{S}}_{\mathrm{eff}}/\beta <1$, where ${\overline{S}}_{\mathrm{eff}}$ is the mean total service requirement per customer. As in classical linear-retrial systems [21], the retrial rate $\eta$ does not enter the stability condition: it governs how fast the orbit is drained once the server is available, not the server’s capacity. As $N\to \infty$, the blocking probability ${\mathsf{\Pi }}_{\mathrm{blk}}\to 0$ and the model approaches the corresponding infinite-capacity retrial queue.

4. Performance Measures

All performance characteristics of the model are derived from the two families of probabilities obtained in Theorem 1: the arbitrary-epoch (time-average) probabilities $P_{j,r}$ and the pre-arrival probabilities $P_{j,r}^{-}$. Two principles are used throughout. (P1) Rate = probability × rate: the steady-state rate of any event listed in Table 1 is obtained by summing its conditional rate over all states in which it may occur, weighted by the corresponding arbitrary-epoch probability. (P2) Arrival-based quantities use pre-arrival probabilities: since the input is general (GI), the state seen by an arrival has distribution $\left\{P_{j,r}^{-}\right\}$, not the time-average $\left\{P_{j,r}\right\}$; the two coincide only for Poisson arrivals (PASTA, [27]). The section is organised into four subsections—server-state and blocking probabilities (Section 4.1), mean system contents (Section 4.2), event rates and flows (Section 4.3), and waiting/sojourn times via Little’s law [28,29] (Section 4.4)—and closes with a flow-balance identity used as an internal correctness check; Table 3 summarises all measures.

4.1. Server-State and Blocking Probabilities

The fractions of time spent in the four server modes are

$$\begin{array}{cccc}\hfill {\mathsf{\Pi }}_0& ={\displaystyle \sum _{r=0}^N}{P}_{0,r},\hfill & \hfill {\mathsf{\Pi }}_1& ={\displaystyle \sum _{r=0}^{N-1}}{P}_{1,r},\hfill \\ \hfill {\mathsf{\Pi }}_2& ={\displaystyle \sum _{r=0}^{N-1}}{P}_{2,r},\hfill & \hfill {\mathsf{\Pi }}_3& ={\displaystyle \sum _{r=1}^N}{P}_{3,r},\hfill \end{array}$$

(13)

and, by the normalisation (12), ${\mathsf{\Pi }}_0+{\mathsf{\Pi }}_1+{\mathsf{\Pi }}_2+{\mathsf{\Pi }}_3=1$. For convenience in the numerical sections we also use the aliases

$${\mathsf{\Pi }}_I={\mathsf{\Pi }}_0+{\mathsf{\Pi }}_3(\mathrm{total}\mathrm{idle}\mathrm{fraction}),{\mathsf{\Pi }}_W={\mathsf{\Pi }}_1(\mathrm{busy}\mathrm{in}\mathrm{vacation}),{\mathsf{\Pi }}_B={\mathsf{\Pi }}_2(\mathrm{busy}\mathrm{in}\mathrm{normal}\mathrm{mode}),$$

(14)

so that the server is busy a fraction ${\mathsf{\Pi }}_1+{\mathsf{\Pi }}_2={\mathsf{\Pi }}_W+{\mathsf{\Pi }}_B$ of the time.

An arriving customer is blocked (lost without joining the orbit) exactly when it finds the system full, i.e., when it sees one of the states $(0,N)$, $(3,N)$ (idle, full orbit) or $(1,N-1)$, $(2,N-1)$ (busy, full orbit). By (P2),

$${\mathsf{\Pi }}_{\mathrm{blk}}=P_{0,N}^{-}+P_{3,N}^{-}+P_{1,N-1}^{-}+P_{2,N-1}^{-}.$$

(15)

Remark 1.

Formula (15) uses pre-arrival probabilities because the input is GI; replacing $P_{j,r}^{-}$ by $P_{j,r}$ is valid only for Poisson arrivals, where $P_{j,r}^{-}=P_{j,r}$ by PASTA [27]. For deterministic arrivals, for instance, the two distributions differ appreciably and the time-average version would bias ${\mathsf{\Pi }}_{\mathrm{blk}}$; all blocking probabilities reported in Section 6 therefore follow (15).

In addition to blocked arrivals, an accepted arrival that finds a busy server may balk: an arrival seeing $(1,r)$ or $(2,r)$, $0\le r\le N-2$, joins the orbit with probability ${\theta }_{r+1}$ and balks with probability ${\overline{\theta }}_{r+1}$; in $(1,N-1)$ and $(2,N-1)$ it is blocked with probability one. Summing over all arrival epochs, the mean loss rate of primary customers and the effective (accepted) arrival rate are

$$B_r=\lambda \left[{\displaystyle \sum _{r=0}^{N-2}}{\overline{\theta }}_{r+1}\left(P_{1,r}^{-}+P_{2,r}^{-}\right)+P_{1,N-1}^{-}+P_{2,N-1}^{-}+P_{0,N}^{-}+P_{3,N}^{-}\right],{\lambda }_{\mathrm{eff}}=\lambda -B_r.$$

(16)

4.2. Mean Orbit Size and Mean System Contents

With all index ranges explicit, the mean orbit size and the mean number of customers in the system are

$$\begin{array}{cc}\hfill L_o& ={\displaystyle \sum _{r=0}^N}r{P}_{0,r}+{\displaystyle \sum _{r=1}^N}r{P}_{3,r}+{\displaystyle \sum _{r=0}^{N-1}}r\left(P_{1,r}+P_{2,r}\right),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill L_s& =L_o+{\mathsf{\Pi }}_1+{\mathsf{\Pi }}_2,\hfill \end{array}$$

(18)

where ${\mathsf{\Pi }}_1+{\mathsf{\Pi }}_2$ is the mean number of customers in service.

4.3. Event Rates and Flows

Applying (P1) to the transition table yields the successful-retrial rate

$$R_{sr}={\displaystyle \sum _{r=1}^N}r\eta \left(P_{0,r}+P_{3,r}\right),$$

(19)

and the reneging and retention rates: each of the (on average) $L_o$ orbiting customers carries a patience clock of rate $\xi$, so clock rings occur at mean rate $\xi L_o$, and each ring leads to abandonment with probability $\alpha$ and to retention with probability $\overline{\alpha }$,

$$R_r=\alpha \xi L_o,R_e=\overline{\alpha }\xi L_o.$$

(20)

The mean service completion rates in vacation mode and in normal mode are obtained by weighting the state-dependent rates ${\nu }_{r+1}$, ${\mu }_{r+1}$ by the corresponding busy probabilities:

$${\mathsf{\Gamma }}_v={\displaystyle \sum _{r=0}^{N-1}}{\nu }_{r+1}{P}_{1,r},{\mathsf{\Gamma }}_b={\displaystyle \sum _{r=0}^{N-1}}{\mu }_{r+1}{P}_{2,r},\mathsf{\Gamma }={\mathsf{\Gamma }}_v+{\mathsf{\Gamma }}_b.$$

(21)

Each completion results in a permanent departure with probability $\beta$, or in a feedback (re-entry into the orbit) with probability $\overline{\beta }$; hence the throughput and the feedback rate are

$$\tau =\beta \mathsf{\Gamma }\left(\mathrm{throughput}\right),F=\overline{\beta }\mathsf{\Gamma }(\mathrm{feedback}\mathrm{rate}).$$

(22)

These are exactly the flow quantities entering the cost and profit functions of Section 5. (The completion rates are denoted ${\mathsf{\Gamma }}_v,{\mathsf{\Gamma }}_b,\mathsf{\Gamma }$ in order to avoid any notational clash with the cost parameters $C_b,C_B$ of Section 5.)

4.4. Waiting and Sojourn Times via Little’s Law

Little’s law $L=\lambda W$ [28,29] applied to the orbit gives the mean time spent in orbit per orbit visit. Orbit departures occur at rate $R_{sr}+R_r$ (successful retrials and reneging; retention events leave the customer in the orbit and are therefore internal), whence

$$W_o={\displaystyle \frac{L_o}{R_{sr}+R_r}}.$$

(23)

Applied to the whole system, whose entries are the accepted primary arrivals (rate ${\lambda }_{\mathrm{eff}}$) and whose exits are the permanent departures (rate $\tau$) and the reneging events (rate $R_r$), we obtain the mean total sojourn time of an accepted primary customer, counting all of its feedback rounds and orbit waits:

$$W_s={\displaystyle \frac{L_s}{{\lambda }_{\mathrm{eff}}}}.$$

(24)

Finally, service visits are initiated at rate ${\lambda }_{\mathrm{eff}}+F$ (primary entries plus feedback re-entries), so that

$$W_s^{\left(\mathrm{visit}\right)}={\displaystyle \frac{L_s}{{\lambda }_{\mathrm{eff}}+F}}$$

(25)

is the mean sojourn time per service visit.

4.5. A Flow-Balance Identity as an Internal Correctness Check

A feedback customer moves from the service position to the orbit without changing the total system occupancy n; feedback is therefore internal at the system level and does not enter the external balance. In steady state, entries into the system (accepted primaries, rate ${\lambda }_{\mathrm{eff}}$) must equal its exits (permanent departures $\tau$ plus reneging $R_r$):

$${\lambda }_{\mathrm{eff}}=\tau +R_r.$$

(26)

Identity (26) is independent of the arrival distribution and involves only the computed probabilities; it therefore provides a demanding, assumption-free check of the numerical solution. In every scenario of Section 6 it holds to within ${10}^{-9}$–${10}^{-15}$ (relative error).

Table 3 summarises the measures for reference.

5. Economic model and optimization

5.1. Cost Elements

With cost rates $C_I$ (idle server), $C_{WV}$ (busy in vacation), $C_B$ (busy, normal), $C_h$ (per customer in the system), $C_o$ (orbit-specific cost per orbiting customer), $C_b$ (per lost/balked customer), $C_r$ (per reneged customer), $C_{re}$ (per retention event), $C_s$ (per completed service), $C_f$ (per feedback event), $C_N$ (per unit of provisioned capacity N), and revenue R per served customer:

$$\begin{array}{cc}\hfill TC& =C_I({\mathsf{\Pi }}_0+{\mathsf{\Pi }}_3)+C_{WV}{\mathsf{\Pi }}_1+C_B{\mathsf{\Pi }}_2+C_h{L}_s+C_o{L}_o+C_b{B}_r+C_r{R}_r+C_{re}{R}_e+C_s\tau +C_{f}F+C_{N}N,\hfill \\ \hfill TR& =R\tau ,TP=TR-TC.\hfill \end{array}$$

Note that $C_h{L}_s+C_o{L}_o$ does not double count: an orbiting customer costs $C_h+C_o$ and a customer in service costs $C_h$. The capacity term $C_{N}N$ represents the cost of provisioning and maintaining a system of capacity N (e.g., buffer, memory, or staffing infrastructure); without it, $TP\left(N\right)$ increases without bound as N grows, since a larger capacity only reduces blocking and raises throughput with no offsetting cost, leaving the optimization over N ill-posed. Table 5 reports the value used, $C_N=3$.

5.2. Optimization Problem and PSO

We maximize $TP(N,\varphi )$ over integer $N\in [1,N_{max}]$ and $\varphi \in [{\varphi }_{min},{\varphi }_{max}]$. Particle positions are rounded for N. The velocity/position updates are

$$\begin{array}{cc}\hfill v_i(t+1)& =w{v}_i\left(t\right)+c_1{r}_1\left({\mathit{pbest}}_i-x_i\left(t\right)\right)+c_2{r}_2\left(\mathit{gbest}-x_i\left(t\right)\right),\hfill \\ \hfill x_i(t+1)& =x_i\left(t\right)+v_i(t+1),\hfill \end{array}$$

with swarm size $P=30$–50, iterations $T_{max}=100$–200, $w=0.7$, $c_1=c_2=1.5$–$2.0$. The complexity per iteration is $O\left(P\right)$ evaluations of Algorithm 1, each $O\left(N^3\right)$.

5.3. Linear Versus Constant Retrial Policy

The classical constant retrial policy replaces the orbit-wide attempt rate $r\eta$ by a constant $\eta$ (see [21]); it is a legitimate alternative model, e.g., for token-based retrial control. Its state-level rate error relative to the linear policy is $\epsilon \left(r\right)=1-1/r$ (50% at $r=2$, 83.3% at $r=6$), which understates orbit departures for $r\ge 2$ and hence overstates orbit occupancy. Section 6 quantifies the opportunity loss incurred when the system is optimized under the constant policy but actually operates under the linear one; the reported gap is a numerical finding of that study rather than a theorem.

6. Numerical Results and Discussion

6.1. Parameter Setting

Table 4 and Table 5 give the baseline parameters (${\theta }_n=1-n/N$, ${\mu }_n=1.2n$, ${\nu }_n=0.8n$).

6.2. Validation

As an internal correctness check independent of any external reference, we validate the analytical solution (Algorithm 1) against an independently coded discrete-event simulation of the same system (Poisson arrivals, $3\times {10}^5$ time units, $5\times {10}^3$ warm-up), sharing no code with the matrix/LST machinery of Section 3. Table 6 reports the agreement at the baseline parameters of Table 4.

6.3. Effect of Arrival Distributions, Retrial Rate, Impatience, and Feedback

Table 7 and Table 8 and Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5 report the effects of various parameters.

6.4. Optimization and Algorithm Comparison

Table 9, Table 10 and Table 11 summarize the PSO results, the comparison with GA and SA, and the robustness analysis, using the cost function of Section 5 with $C_N=3$.

7. Conclusions

We have analyzed a GI/M/1/N retrial queue with multiple working vacations, Bernoulli feedback, and impatient customers with retention, under the linear retrial policy in which the total attempt rate is $r\eta$. The contributions are: a consistent four-mode formulation; complete steady-state distributions via the supplementary variable technique and the embedded arrival-epoch chain, with closed-form matrix exponentials for D, $E_k$, and $H_2$ inputs; a degenerate-limit result connecting the model, as $\eta \to \infty$, to the waiting-line model of [19], with Algorithm 1 itself validated numerically against an independent discrete-event simulation; corrected performance measures with an internal flow-balance check; stability remarks; and an economic model optimized by PSO with GA/SA comparisons and cost robustness analysis. Future work includes multi-server retrial queues, impatience during service, state-dependent retrial rates, correlated (MAP) arrivals, alternative vacation trigger policies, and machine-learning approaches for real-time parameter tuning; applications to network slicing in 6G are a natural outlet.