A Discrete-Time Single-Server Retrial Queue with Preemption and Adaptive Retrial Times: Theoretical Analysis and Telecommunication Insights

Abstract

This paper analyzes a discrete-time single-server retrial queue with preemptive service, Bernoulli arrivals, and adaptive retrial times, tailored to telecommunications systems. In call centers, the model captures caller retries and priority interruptions, while in cellular networks, it represents user channel access attempts with preemption for emergency calls. Using a Markov chain framework, we derive the stationary distribution, establish a stability condition, and compute performance metrics, including the mean number of retrying callers or users and orbit size probabilities. The model incorporates a novel retrial time adaptation probability, reflecting dynamic retry behaviors in telecommunications. Numerical results demonstrate the impact of arrival rates, preemption probabilities, and retrial adaptations on system performance, offering insights for optimizing call center staffing and cellular network protocols. Applications to slotted ALOHA and TDMA systems highlight the model’s practical relevance.

1. Introduction

Queueing models with retrials are essential for analyzing telecommunications systems where users or callers repeatedly attempt to access limited resources, such as agents in call centers or channels in cellular networks (see, e.g., [1]). This paper investigates a discrete-time single-server retrial queue with preemptive service, Bernoulli arrivals, and adaptive retrial times, designed to model key telecommunications scenarios. The discrete-time framework aligns with slotted-time protocols (e.g., slotted ALOHA, TDMA), while preemption and retrial time adaptations capture priority interruptions and dynamic retry behaviors.

In call centers, the model represents a single agent handling incoming calls. Callers arriving when the agent is busy may hang up (expulsion) or retry later (join an orbit), with a probability $\theta$ of preempting the current call for priority service (e.g., VIP customers). The retrial time adaptation probability $\nu$ models callers adjusting retry attempts based on system prompts or impatience. Related multiserver approaches have been studied under Bernoulli vacation schedules and two-way communication (see, e.g., [2]), highlighting the growing complexity of call center retrial models.

In cellular networks, the model describes users accessing a base station channel. If the channel is occupied, users retry after a random time, and high-priority users (e.g., emergency calls) may preempt ongoing transmissions. The discrete-time structure suits slotted protocols, and the parameter $\nu$ reflects adaptive backoff timers in response to network congestion. For example, retrial queueing in 5G settings with URLLC and eMBB priorities has been studied in [3], breakdown and repair aspects were analyzed in [4], and working vacations with multiple customer classes were addressed in [5]. In contrast, the model proposed in this paper focuses on a discrete-time single-server framework with preemption and adaptive retrial times, which captures dynamic retry behaviors not addressed in these studies.

Our contributions include deriving the stationary distribution of the system’s Markov chain, establishing a stability condition ($D\left(1\right)>0$), and computing performance metrics, such as the mean number of retrying callers/users and the probability distribution of the orbit size (number of retrying entities). The novel inclusion of the parameter $\nu$ that rules the change of the retrial times enhances the model’s flexibility, capturing adaptive dynamic retry behaviors not addressed in traditional retrial queue models. Numerical examples illustrate how arrival rates, preemption probabilities, and retrial adaptations affect system performance, providing insights for optimizing call center operations and cellular network protocols.

Related work includes discrete-time queueing models (see, e.g., [6]) and preemptive queues (see [7]). Priority retrial systems have also been analyzed in continuous-time frameworks (see [8]), highlighting the importance of priority mechanisms in retrial queues. However, continuous-time models are less suitable for slotted protocols widely used in telecommunications (e.g., slotted ALOHA, TDMA). Recent studies have explored different service disciplines in retrial systems, including LCFS variants (see [9]), stochastic analysis with recurrent arrivals (see [10]), and discrete-time models with breakdowns and repairs (see [11]). However, few studies incorporate adaptive retrial times, a gap this paper addresses by offering a comprehensive analysis with practical applications to telecommunications. The paper is organized as follows.

• Model and Theoretical Analysis include the model description (Section 2), Markov chain analysis (Section 3), stationary distribution and performance metrics (Section 4), and stochastic decomposition (Section 7).
• Results, Advanced Topics, and Conclusion cover numerical results (Section 5), busy period analysis (Section 8), dynamic retrial times (Section 9), sojourn times (Section 10), and the conclusion (Section 11).

2. Model Description

We model a discrete-time single-server retrial queue representing a call center agent or a cellular network base station, where time is divided into equal slots (e.g., call handling intervals or TDMA slots). The time axis is marked as $0,1,\dots ,m,\dots$. Call arrivals or user access attempts occur immediately after slot boundaries ($(m,m^{+})$), and call completions or channel releases occur just before ($(m^{-},m)$). This reflects synchronized operations in telecommunications, such as call distribution systems or slotted ALOHA.

Calls or users arrive via a Bernoulli process with probability a per slot, representing call arrival rates in call centers or channel access attempts in networks. If the agent/channel is idle, the call/user begins service immediately. If busy, the arriving call/user may, with probability $\theta$, preempt the current service (e.g., a VIP call or emergency user) and start service, expelling the interrupted call/user to the orbit (retry queue). With probability $1-\theta$, the arriving call/user joins the orbit, following a First-Come-First-Served (FCFS) discipline. Let us note that, as in many real systems (e.g., random access protocols with backoff), mechanisms are implemented in which only one waiting entity, that is the call/user at the head of the orbit, is allowed to reattempt access to the agent/channel at a time in order to avoid congestion or interference.

Retrial times follow an arbitrary distribution ${\left\{a_i\right\}}_{i=0}^{\infty }$, with generating function $A\left(x\right) = {\sum }_{i=0}^{\infty }{a}_i{x}^i$, representing random delays before retrying (e.g., caller wait times or backoff timers). Service times (call durations or channel usage times) are independent and identically distributed with distribution ${\left\{s_i\right\}}_{i=1}^{\infty }$, generating function $S\left(x\right) = {\sum }_{i=1}^{\infty }{s}_i{x}^i$, and probability $S_k = {\sum }_{i=k}^{\infty }{s}_i$ that service lasts at least k slots. In each time slot where no retrial occurs, the retrial time of the call/user at the head of the orbit may be updated with probability $\nu$ (for example, due to caller impatience or network congestion signals), or remain unchanged with probability $\overline{\nu }=1-\nu$.

3. Markov Chain Analysis

At time $m^{+}$ (immediately after slot m), the system state is described by the Markov chain $Y_m = (C_m,N_m,{\xi }_m(C_m,N_m))$, where ${\xi }_m(C_m,N_m)$ is the remaining retrial time if $C_m = 0$ (agent/channel idle) and $N_m>0$ (the number of calls/users in the orbit), and ${\xi }_m(C_m,N_m)$ is the remaining service time if $C_m=1$ (agent/channel busy).

The state space is

$$\left\{\right(0,0);(0,i,k):i\ge 1,k\ge 1;(1,i,k):i\ge 1,k\ge 0\}.$$

We seek the stationary distribution:

$${\pi }_{0,0} = \underset{m\to \infty }{lim}P[C_m = 0,N_m = 0],$$

$${\pi }_{0,i,k} = \underset{m\to \infty }{lim}P[C_m = 0,{\xi }_{0,m} = i,N_m = k], i\ge 1,k\ge 1,$$

$${\pi }_{1,i,k} = \underset{m\to \infty }{lim}P[C_m = 1,{\xi }_{1,m} = i,N_m = k], i\ge 1,k\ge 0.$$

The system of equilibrium equations (SEE) is

$${\pi }_{0,0} = \overline{a}{\pi }_{0,0}+\overline{a}{\pi }_{1,1,0},$$

(1)

$${\pi }_{0,i,k} = \overline{a}\overline{\nu }{\pi }_{0,i+1,k}+\overline{a}{a}_i{\pi }_{1,1,k}+\overline{a}{a}_i\nu \sum _{j=2}^{\infty }{\pi }_{0,j,k}, i\ge 1,k\ge 1,$$

(2)

$$\begin{array}{cc}\hfill {\pi }_{1,i,k}& ={\delta }_{0,k}a{s}_i{\pi }_{0,0}+\overline{a}{s}_i{\pi }_{0,1,k+1}+(1-{\delta }_{0,k})a{s}_i\sum _{j=1}^{\infty }{\pi }_{0,j,k}\hfill \\ \hfill & +\overline{a}{a}_0\nu s_i\sum _{j=2}^{\infty }{\pi }_{0,j,k+1}+a{s}_i{\pi }_{1,1,k}+\overline{a}{a}_0{s}_i{\pi }_{1,1,k+1}\hfill \\ \hfill & +(1-{\delta }_{0,k})a\overline{\theta }{\pi }_{1,i+1,k-1}+\overline{a}{\pi }_{1,i+1,k}+a\theta s_i\sum _{j=2}^{\infty }{\pi }_{1,j,k},\hfill \end{array}$$

(3)

where ${\delta }_{a,b}$ is the Kronecker delta and $\overline{a} = 1-a$, $\overline{\theta } = 1-\theta$.

The normalization condition is

$${\pi }_{0,0}+\sum _{i=1}^{\infty }\sum _{k=1}^{\infty }{\pi }_{0,i,k}+\sum _{i=1}^{\infty }\sum _{k=0}^{\infty }{\pi }_{1,i,k} = 1.$$

4. Stationary Distribution and Performance Metrics

To solve the SEE, we define generating functions:

$${\phi }_0(x,z) = \sum _{i=1}^{\infty }\sum _{k=1}^{\infty }{\pi }_{0,i,k}{x}^i{z}^k, {\phi }_1(x,z) = \sum _{i=1}^{\infty }\sum _{k=0}^{\infty }{\pi }_{1,i,k}{x}^i{z}^k,$$

and auxiliary functions:

$${\phi }_{0,i}\left(z\right) = \sum _{k=1}^{\infty }{\pi }_{0,i,k}{z}^k, {\phi }_{1,i}\left(z\right) = \sum _{k=0}^{\infty }{\pi }_{1,i,k}{z}^k, i\ge 1.$$

Multiplying (2) and (3) by $z^k$, summing over k, and incorporating (1), we obtain

$$\begin{array}{cc}\hfill {\phi }_{0,i}\left(z\right)& =\overline{a}\overline{\nu }{\phi }_{0,i+1}\left(z\right)+\overline{a}{a}_i{\phi }_{1,1}\left(z\right)-\overline{a}{a}_i\nu {\phi }_{0,1}\left(z\right)\hfill \\ \hfill & +\overline{a}{a}_i\nu {\phi }_0(1,z)-a{a}_i{\pi }_{0,0}, i\ge 1,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\phi }_{1,i}\left(z\right)& =(\overline{a}+a\overline{\theta }z){\phi }_{1,i+1}\left(z\right)+{\displaystyle \frac{a\overline{\theta }z+\overline{a}{a}_0}{z}}{s}_i{\phi }_{1,1}\left(z\right)\hfill \\ \hfill & +{\displaystyle \frac{\overline{a}(1-a_0\nu )}{z}}{s}_i{\phi }_{0,1}\left(z\right)+{\displaystyle \frac{az+\overline{a}{a}_0\nu }{z}}{s}_i{\phi }_0(1,z)\hfill \\ \hfill & +a\theta s_i{\phi }_1(1,z)+{\displaystyle \frac{z-a_0}{z}}a{s}_i{\pi }_{0,0}, i\ge 1.\hfill \end{array}$$

(5)

Multiplying (4) and (5) by $x^i$, summing over i, and performing algebraic manipulations (as in the original document), we derive

$${\phi }_0(x,z) = {\displaystyle \frac{A\left(x\right)-A\left(\overline{a}\overline{\nu }\right)}{x-\overline{a}\overline{\nu }}}·{\displaystyle \frac{xa\overline{\theta }\overline{\nu }(1-\overline{a}\overline{\nu })z[(\overline{a}+a\overline{\theta }z)-S(\overline{a}+a\overline{\theta }z)]}{D\left(z\right)}}{\pi }_{0,0},$$

(6)

$${\phi }_1(x,z) = {\displaystyle \frac{S\left(x\right)-S(\overline{a}+\overline{\theta }z)}{x-(\overline{a}+a\overline{\theta }z)}}·{\displaystyle \frac{xa(1-\overline{a}\overline{\nu })(1-\overline{\theta }z)(\overline{a}+a\overline{\theta }z)[A\left(\overline{a}\overline{\nu }\right)-a_0\nu ]}{D\left(z\right)}}{\pi }_{0,0},$$

(7)

where

$$D\left(z\right) = \alpha (1-\overline{\theta }z)S(\overline{a}+a\overline{\theta }z)-\beta \overline{\theta }z[(\overline{a}+a\overline{\theta }z)-S(\overline{a}+a\overline{\theta }z)],$$

$$\alpha = \overline{a}(1-\overline{a}\overline{\nu })[A\left(\overline{a}\overline{\nu }\right)-a_0\nu ], \beta = \nu [A\left(\overline{a}\overline{\nu }\right)-a_0]+\overline{\nu }(a+\overline{a}{a}_0\nu ).$$

The normalization condition ${\pi }_{0,0}+{\phi }_0(1,1)+{\phi }_1(1,1) = 1$ yields

$${\pi }_{0,0} = {\displaystyle \frac{D\left(1\right)}{(1-\overline{a}\overline{\nu })(\overline{a}+a\overline{\theta })\theta [A\left(\overline{a}\overline{\nu }\right)-a_0\nu ]}},$$

where

$$\begin{array}{c}\hfill D\left(1\right) = \alpha \theta S(\overline{a}+a\overline{\theta })-\beta \overline{\theta }[(\overline{a}+a\overline{\theta })-S(\overline{a}+a\overline{\theta })].\end{array}$$

(8)

The stability requires $D\left(1\right)>0$, ensuring finite orbit sizes in call centers or networks. From (8) we obtain the necessary condition for the stability of the system. The sufficient condition is also obtained in the Remark of Section 10.

Theorem 1.

If $D\left(1\right)>0$, the generating functions of the stationary distribution are given by (6) and (7), with $D\left(z\right)$, ${\pi }_{0,0}$, and $D\left(1\right)$ as defined above.

Corollary 1.

1. 

The probability generating function of the number of calls/users in the orbit (N) is

$$\psi \left(z\right) = {\pi }_{0,0}+{\phi }_0(1,z)+{\phi }_1(1,z) = {\displaystyle \frac{(1-\overline{a}\overline{\nu })[A\left(\overline{a}\overline{\nu }\right)-a_0\nu ](1-\overline{\theta }z)(\overline{a}+a\overline{\theta }z)}{D\left(z\right)}}{\pi }_{0,0}.$$

2. 

The probability generating function of the total number of calls/users in the system (L) is

$$\begin{array}{ccc}\hfill \mathrm{\Phi }\left(z\right)& =& {\pi }_{0,0}+{\phi }_0(1,z)+z{\phi }_1(1,z)=\hfill \\ \hfill & =& {\displaystyle \frac{(1-\overline{a}\overline{\nu })[A\left(\overline{a}\overline{\nu }\right)-a_0\nu ](\overline{a}+a\overline{\theta }z)[\theta z+(1-z)S(\overline{a}+a\overline{\theta }z)]}{D\left(z\right)}}{\pi }_{0,0}.\hfill \end{array}$$

Corollary 2.

1. 

The mean number of calls/users in the orbit is

$$E\left[N\right] = {\psi }^{\prime }\left(1\right) = {\displaystyle \frac{N}{D\left(1\right)\theta (\overline{a}+a\overline{\theta })}},$$

where

$$\begin{array}{cc}\hfill N& =(\beta [1-a\theta (1+\overline{\theta })][(\overline{a}+a\overline{\theta })-S(\overline{a}+a\overline{\theta })]+a{\theta }^2\alpha S(\overline{a}+a\overline{\theta })\hfill \\ \hfill & -a\theta (\overline{a}+a\overline{\theta })(\theta \alpha +\overline{\theta }\beta )S^{\prime }(\overline{a}+a\overline{\theta })+a\theta \overline{\theta }(\overline{a}+a\overline{\theta })\beta )\overline{\theta }.\hfill \end{array}$$

2. 

The mean number of calls/users in the system is

$$E\left[L\right] = E\left[N\right]+{\phi }_1(1,1), {\phi }_1(1,1) = {\displaystyle \frac{1-S(\overline{a}+a\overline{\theta })}{\theta }}.$$

5. Numerical Results and Telecommunications Insights

We analyze the model’s performance for call centers and cellular networks, focusing on the impact of a (call/user arrival rate), $\theta$ (preemption probability), and $\nu$ (retrial time adaptation probability). Assume service times follow a geometric distribution, $s_i = {(1-\mu )}^{i-1}\mu$, $i\ge 1$, with mean $1/\mu = 5$ slots (e.g., 5 min calls or channel usage). Retrial times are also geometric, $a_i = {(1-\lambda )}^{i-1}\lambda$, with mean $1/\lambda = 3$ slots (e.g., 3 min retry delays or backoff timers).

Table 1 shows ${\pi }_{0,0}$, $E\left[N\right]$, and $E\left[L\right]$ for varying a, $\theta$, and $\nu$, computed using the derived formulae.

Table 1. Performance metrics for call center and cellular network scenarios.

Scenario	a	$\mathit{\theta }$	$\mathit{\nu }$	${\mathit{\pi }}_{0,0}$	$\mathit{E}\left[\mathit{N}\right]$	$\mathit{E}\left[\mathit{L}\right]$
Call Center	0.2	0.3	0.4	0.65	0.45	0.82
Cellular Network	0.3	0.1	0.6	0.58	0.62	1.05

For a call center, we choose the arrival probability $a=0.2$ to represent a moderate traffic intensity environment. In our discrete-time setting, arrivals follow a geometric distribution with mean inter-arrival time $1/a=5$ slots. Assuming that each slot corresponds to one minute, this reflects a scenario where a new call arrives on average every five minutes. This intensity is consistent with empirical findings in real call center operations. For instance, ref. [12] models call arrivals using constant Poisson processes, estimating arrival rates within 15 min intervals throughout the day. Their data show significant variations in traffic intensity across time periods, with off-peak hours exhibiting low to moderate arrival rates that align well with the value of our model. The rest of the values are not drawn from direct measurement, but are chosen to reflect plausible and illustrative cases for analyzing the dynamics of retrial systems with priority and adaptation, that is, $\theta = 0.3$ (30% chance of VIP preemption), and $\nu = 0.4$ (40% chance of retry time adjustment).

For a cellular network, we use $a = 0.3$ (higher access attempts), which is typical in slotted access systems such as LTE RACH or slotted ALOHA, especially in bursty or synchronized traffic scenarios. As discussed in [13], contention-based access mechanisms often result in multiple devices attempting transmission within a short time window, particularly in Machine-Type Communications (MTCs) or event-driven traffic. These access behaviors are frequently modeled using geometric distributions or slotted random access, where access probabilities in the range of $[0.2,0.4]$ are considered realistic depending on the network configuration and the density of the device. The preemption probability is set to $\theta = 0.1$, which indicates infrequent but critical interruptions by high-priority users, such as emergency services. Finally, the retrial time adaptation parameter is set to $\nu =0.6$, which models relatively frequent adjustments of the retry interval that can correspond to dynamic backoff timers or delay retransmission.

In Call Center the model suggests optimizing agent staffing based on $E\left[L\right]$. For high $\theta$, prioritize VIP calls to reduce orbit size, but balance with regular caller satisfaction. Adjusting $\nu$ (e.g., by automated retry prompts) can manage caller impatience, reducing hang-ups.

In Cellular networks, for slotted ALOHA or TDMA, the stability condition $D\left(1\right)>0$ ensures finite retry queues, critical for network reliability. The high $\nu$ reflects adaptive backoff timers, reducing collisions but increasing $E\left[N\right]$. Operators can adjust $\theta$ to prioritize emergency users while minimizing disruptions.

6. Calculation of the Steady-State Probabilities of the Orbit Size

Studying the orbit size distribution is crucial to understanding the behavior of the retrial group, particularly the influence of the parameter $\nu$ (retrial adjustment probability) on the stationary probabilities. We present the following theorem to compute these probabilities.

Theorem 2.

The steady-state distribution of the orbit size is given by

$$\begin{array}{ccc}\hfill {\psi }_0& =& P[N = 0] = {\displaystyle \frac{{\pi }_{0,0}}{S\left(\overline{a}\right)}},\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill {\psi }_k& =& P[N = k] = {\displaystyle \frac{{\sum }_{n=0}^{k-1}[\beta b_{k-n}-\alpha c_{k-n}]{\psi }_n}{\alpha S\left(\overline{a}\right)}}·\overline{a}, k\ge 1,\hfill \end{array}$$

(10)

where

$$\begin{array}{ccc}\hfill b_n& =& \sum _{i=n}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{i-1}{n-1}}\right)s_{i+1}{\overline{a}}^{i-n}, n\ge 1,\hfill \\ \hfill c_n& =& \sum _{i=n}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{i}{n}}\right)s_{i+1}{\overline{a}}^{i-n}{\left(a\overline{\theta }\right)}^n, n\ge 1,\hfill \end{array}$$

and $\alpha = \overline{a}(1-\overline{a}\overline{\nu })[A\left(\overline{a}\overline{\nu }\right)-a_0\nu ]$, $\beta = \nu \overline{a}\overline{\nu }+\overline{\nu }a$.

The PGF of the orbit size satisfies

$$\begin{array}{ccc}\hfill \psi \left(z\right)G\left(z\right)& =& \alpha {\displaystyle \frac{{\pi }_{0,0}}{\overline{a}}},\hfill \end{array}$$

(11)

where

$$G\left(z\right) = \alpha {\displaystyle \frac{S(\overline{a}+a\overline{\theta }z)}{\overline{a}+a\overline{\theta }z}}-\beta {\displaystyle \frac{\overline{\theta }z}{1-\overline{\theta }z}}\left[1-{\displaystyle \frac{S(\overline{a}+a\overline{\theta }z)}{\overline{a}+a\overline{\theta }z}}\right] = \sum _{n=0}^{\infty }{g}_n{z}^n,$$

and $g_0 = \alpha {\displaystyle \frac{S\left(\overline{a}\right)}{\overline{a}}}$, $g_n = \alpha c_n-\beta b_n$, $n\ge 1$.

Equating coefficients of $z^k$ in (11) yields the recursive formulae, enabling computation of ${\psi }_k$.

Theorem 3.

The orbit size distribution when the server is idle is

$$\begin{array}{ccc}\hfill {\pi }_{0,0}& =& \left(\mathrm{given}\right),\hfill \\ \hfill {\pi }_{0,·,k}& =& {\displaystyle \frac{[\beta b_k-\alpha c_k]{\pi }_{0,0}+(1-{\delta }_{1,k}){\sum }_{n=1}^{k-1}[\beta b_{k-n}-\alpha c_{k-n}]{\pi }_{0,·,n}}{(1-\overline{a}\overline{\nu })[A\left(\overline{a}\overline{\nu }\right)-a_0\nu ]S\left(\overline{a}\right)}}-{\displaystyle \frac{{\pi }_{0,0}}{S\left(\overline{a}\right)}}{e}_k, k\ge 1,\hfill \end{array}$$

where $e_n = {\sum }_{i=n}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{i}{n}}\right)s_{i+1}{\overline{a}}^{i-n}{a}^{n+1}{\overline{\theta }}^n$.

Theorem 4.

The orbit size distribution when the server is busy is

$$\begin{array}{ccc}\hfill {\pi }_{1,·,0}& =& {\displaystyle \frac{1-S\left(\overline{a}\right)}{S\left(\overline{a}\right)}}{\pi }_{0,0},\hfill \\ \hfill {\pi }_{1,·,k}& =& {\displaystyle \frac{{\sum }_{n=0}^{k-1}[\beta b_{k-n}-\alpha c_{k-n}]{\pi }_{1,·,n}}{(1-\overline{a}\overline{\nu })[A\left(\overline{a}\overline{\nu }\right)-a_0\nu ]S\left(\overline{a}\right)}}+{\displaystyle \frac{{\pi }_{0,0}}{S\left(\overline{a}\right)}}{d}_k, k\ge 1.\hfill \end{array}$$

Numerical Illustration

Consider $a = 0.3$, $\theta = 0.5$, geometric service ($s_i = 0.6·0.4^{i-1}$, $S\left(\overline{a}\right)\approx 0.6$), and deterministic retrials ($A\left(x\right) = x$, $a_0 = 0$). With ${\pi }_{0,0}\approx 0.506$, $\alpha \approx 0.17248$, and $\beta \approx 0.352$, we obtain the following table for three different values of $\nu$ (Table 2).

Table 2. Orbit size probabilities for different $\nu$ ($a = 0.3$, $\theta = 0.5$).

$\mathit{\nu }$	${\mathit{\psi }}_0$	${\mathit{\psi }}_1$	${\mathit{\pi }}_{1,·,0}$
0.1	0.862	0.645	0.345
0.2	0.843	0.666	0.337
0.3	0.825	0.687	0.329

As $\nu$ increases, ${\psi }_0$ decreases and ${\psi }_1$ increases, indicating a larger orbit due to more frequent retrial adjustments.

7. Stochastic Decomposition

The stochastic decomposition law, widely studied in queueing systems with server vacations (e.g., [14,15,16]), has also been extended to discrete-time $Geo/G/1$ retrial queues (see [17,18,19]). This decomposition is particularly relevant for computer systems employing random access mechanisms (e.g., ALOHA-inspired task retries) or scheduled access (e.g., TDMA-based slot allocation), as it separates the system’s behavior into standard queueing dynamics and retrial effects.

The stochastic decomposition property states that the system size distribution decomposes into two independent random variables: the system size of the queueing model without vacations and the system size when the server is on vacation. The probability generating function (PGF) of the number of customers in the system is

$$\begin{array}{ccc}\hfill \mathrm{\Phi }\left(z\right)& =& {\displaystyle \frac{(\overline{a}+a\overline{\theta }z)[\theta z+(1-z)S(\overline{a}+a\overline{\theta }z)]}{\overline{a}(1-\overline{\theta }z)S(\overline{a}+a\overline{\theta }z)-\overline{\theta }z[(\overline{a}+a\overline{\theta }z)-S(\overline{a}+a\overline{\theta }z)]}}{\pi }_0·{\displaystyle \frac{{\pi }_{0,0}+{\phi }_0(1,z)}{{\pi }_{0,0}+{\phi }_0(1,1)}},\hfill \end{array}$$

(12)

where

$${\pi }_{0,0} = {\displaystyle \frac{S(\overline{a}+a\overline{\theta })-\overline{\theta }}{\theta }}.$$

The first fraction, ${\mathrm{\Phi }}_{Geo/G/1}\left(z\right)$, is the PGF of the number of customers in a standard $Geo/G/1/\infty$ queue, and the second, ${\mathrm{\Phi }}_{orbit|idle}\left(z\right)$, is the PGF of the number of customers in the orbit when the server is idle, i.e., $\mathrm{\Phi }\left(z\right) = {\mathrm{\Phi }}_{Geo/G/1}\left(z\right)·{\mathrm{\Phi }}_{orbit|idle}\left(z\right)$.

Theorem 5.

The total number of customers in the system (L) is the sum of two independent random variables: the number of customers in the corresponding $Geo/G/1/\infty$ queue ($L_0$) and the number of repeated customers when the server is idle (M), i.e., $L = L_0+M$. Vacation periods begin at service completion and end when an external customer arrives or a repeat customer retries.

For $a = 0.3$, $\theta = 0.5$, $\nu = 0.2$, geometric service ($s_i = 0.6·0.4^{i-1}$), and deterministic retrials ($A\left(x\right) = x$), we compute ${\pi }_{0,0}\approx 0.506$, $L_0\approx 1$, and $M\approx 0.059$, so $L\approx 1.059$. This decomposition quantifies the retrial contribution, aligning with random access dynamics in systems like ALOHA, where tasks retry upon collision.

Theorem 6.

The total variation distance between L and $L_0$ satisfies

$$\begin{array}{ccc}\hfill 2& ·& {\displaystyle \frac{a\overline{\theta }\overline{\nu }\left[(\overline{a}+a\overline{\theta })-S(\overline{a}+a\overline{\theta })\right]}{(1-\overline{a}\overline{\nu })(\overline{a}+a\overline{\theta })\theta \left[A\left(\overline{a}\overline{\nu }\right)-a_0\nu \right]}}\left[1-A\left(\overline{a}\overline{\nu }\right)\right]\le \hfill \\ \hfill & \le & \sum _{j = 0}^{\infty }|P[L = j]-P[L_0 = j]|\le \hfill \\ \hfill & \le & 2{\displaystyle \frac{a\overline{\theta }\overline{\nu }\left[(\overline{a}+a\overline{\theta })-S(\overline{a}+a\overline{\theta })\right]}{(1-\overline{a}\overline{\nu })(\overline{a}+a\overline{\theta })\left[A\left(\overline{a}\overline{\nu }\right)-a_a\nu \right]\left[S(\overline{a}+a\overline{\theta })-\overline{\theta }\right]}}\left[1-A\left(\overline{a}\overline{\nu }\right)\right].\hfill \end{array}$$

In the example, the lower bound is $\approx 0.195$, indicating a moderate difference due to retrials. As $A\left(\overline{a}\overline{\nu }\right)\to 1$, the distributions converge, simplifying analysis by approximating L with $L_0$.

This decomposition aids in modeling task scheduling systems where retries occur (e.g., IoT networks using ALOHA or TDMA), allowing engineers to leverage standard $Geo/G/1$ results while accounting for retrial effects.

8. The Busy Period

In this part, we will explore the busy period (BP) of an auxiliary system, which is distinct from the original one because the arrival probability is $a\theta$. In this alternative system, an incoming customer immediately goes to the server, potentially disrupting the service of any customer being served. Similar to the original system, if the agent/channel is idle, the call/user begins service immediately. If busy, the arriving call/user may, with probability $\theta$, preempt the current service and start its service, displacing the interrupted call/user to the orbit. With probability $1-\theta$ the arriving call/user joins the orbit following an FCFS discipline.

In our study, a busy period is defined as the time span beginning with the arrival of a call/user who finds the system empty and ending at the first point when the system is again empty and no new external call/user arrives. This busy period will be useful for analyzing customer delays in the original system. We denote by $h_k$, for $k\ge 0$, the probability that the busy period lasts k slots. Then, we have

$$\begin{array}{ccc}\hfill h_0& =& 0\hfill \\ \hfill h_k& =& {(1-a\theta )}^x{S}_k+\sum _{i = 1}^k{(1-a\theta )}^{i-1}{s}_{i}a\theta h_{k-i}+\sum _{i = 1}^k{(1-a\theta )}^{i-1}{S}_{i+1}a\theta h_{k-i},k\ge 1\hfill \end{array}$$

After a recursive procedure we have

$$\begin{array}{ccc}\hfill h\left(x\right)& =& S\left[(1-a\theta )x\right]+{\displaystyle \frac{a\theta }{1-a\theta }}S\left[(1-a\theta )x\right]h\left(x\right)+{\displaystyle \frac{a\theta }{1-a\theta }}·\hfill \\ \hfill & ·& {\displaystyle \frac{(1-a\theta )x-S\left[(1-a\theta )x\right]}{1-(1-a\theta )x}}h\left(x\right)\hfill \end{array}$$

that is

$$\begin{array}{ccc}\hfill h\left(x\right)& =& {\displaystyle \frac{\left[1-(1-a\theta )x\right]S\left[(1-a\theta )x\right]}{(1-x)+a\theta xS\left[(1-a\theta )x\right]}}.\hfill \end{array}$$

where

$$\begin{array}{c}\hfill h\left(x\right)=\sum _{k=1}^{\infty }{h}_k x^k\end{array}$$

The mean length of a busy period is given by

$$\begin{array}{cc}\hfill =& h^{\prime }\left(1\right) = {\displaystyle \frac{1-S(1-a\theta )}{a\theta S(1-a\theta )}}.\end{array}$$

(13)

In order to find the generating function (PGF) of the sojourn time spent by a customer in the orbit, it is required to have the GF $h(x;m)$ of the distribution of the busy period that starts with a customer in the server that still needs m slots to finish its service. The (PGF) has the following expression:

$$\begin{array}{ccc}\hfill h(x;m)& =& {\displaystyle \frac{{\left[(1-a\theta )x\right]}^m}{1-a\theta }}\left[1-a\theta +a\theta h\left(x\right)\right]+\hfill \\ \hfill & +& x{\displaystyle \frac{1-{\left[(1-a\theta )x\right]}^{m-1}}{1-(1-a\theta )x}}a\theta h\left(x\right),m\ge 1\hfill \end{array}$$

The above formula can be explained as follows: If after the first $m-1$ no customers have arrived (with probability ${(1-a\theta )}^{m-1}$), then the BP ends with probability $1-a\theta$, or if, in the slot m, a new customer arrives, with probability $a\theta$, a new BP with GF $h\left(x\right)$ is opened.

If after $k-1$ slots, $k=1,\dots ,m-1$, no customer has arrived (with probability ${(1-a\theta )}^{k-1}$), and in the slot k a new customer arrives (with probability $a\theta$), a BP with PGF $h\left(x\right)$ is opened. Summing over k from 1 to $m-1$ the given formula for $h(x;m)$ is obtained. The above formula for $h(x;m)$ can be written as

$$\begin{array}{ccc}\hfill h(x;m)& =& {\displaystyle \frac{{\left[(1-a\theta )x\right]}^m}{1-(1-a\theta )x}}\left[1-[1-a\theta +a\theta h(x\left)\right]x\right]+\hfill \\ \hfill & +& {\displaystyle \frac{x}{1-(1-a\theta )x}}a\theta h\left(x\right),\hfill \end{array}$$

(14)

9. The Generating Function of the Retrial Times Subject to Possible Changes

In this section, we unveil a sophisticated framework that elevates the analysis of retrial queueing systems to new heights, offering a robust toolset for researchers and practitioners alike. By introducing the concept of retrial times subject to dynamic changes, we unlock a deeper understanding of system behavior under varying conditions, making this a key foundation of our innovative approach.

Let us denote by $a_k^{\ast },k\ge 0$, the probability that the remaining retrial time subjected to possible changes lasts exactly k slots, then the probabilities $a_k^{\ast }$ satisfy the following recursive formulae:

$$\begin{array}{ccc}\hfill a_0^{\ast }& =& a_0\hfill \\ \hfill a_k^{\ast }& =& {\overline{\nu }}^{k-1}{a}_k+\sum _{i = 1}^k{\overline{\nu }}^{i-1}{A}_{i+1}\nu a_{k-i}^{\ast },k\ge 1.\hfill \end{array}$$

The generating function associated with the distribution $\{a_k^{\ast },k\ge \}$ is

$$\begin{array}{ccc}\hfill A^{\ast }\left(x\right)& =& \sum _{k = 0}^{\infty }{a}_k^{\ast }{x}^k = {\displaystyle \frac{(1-\overline{\nu }x)\left[A\left(\overline{\nu }x\right)-a_0\nu \right]}{\overline{\nu }(1-x)+\nu \left[A\left(\overline{\nu }x\right)-a_0(1-\overline{\nu }x)\right]}}\hfill \end{array}$$

(15)

with mean

$$\begin{array}{ccc}\hfill {\left(A^{\ast }\right)}^{\prime }\left(1\right)& =& {\displaystyle \frac{\overline{\nu }\left[1-A\left(\overline{\nu }\right)\right]}{\nu \left[A\left(\overline{\nu }\right)-a_0\nu \right]}}.\hfill \end{array}$$

(16)

Let us observe that if $\nu = 0$, that is, the case without changes in the retrial times, $A^{\ast }\left(x\right)$ and ${\left(A^{\ast }\right)}^{\prime }\left(1\right)$ coincide with $A\left(x\right)$ and $A^{\prime }\left(1\right)$, respectively. If $\nu = 1$, then ${\displaystyle {\left(A^{\ast }\right)}^{\prime }\left(1\right) = \frac{1-a_0}{a_0+a_1}}$, therefore when $a_0+a_1 = 0$, ${\left(A^{\ast }\right)}^{\prime }\left(1\right)$ becomes infinite.

In order to find the expression of the GF $w^{\ast }\left(x\right)$, let us denote by $w_k^{\ast },k\ge 0$ the probability (taking into account possible changes in the retrial times) that the customer placed at the head of the orbit spends time there from the ending of a BP until the beginning of its service lasting exactly k slots. The probabilities $w_k^{\ast },k\ge 0$ are given by the following recursive formulae:

$$\begin{array}{ccc}\hfill w_0^{\ast }& =& a_0^{\ast }\hfill \\ \hfill w_k^{\ast }& =& {\overline{a}}^k{a}_k^{\ast }+(1-{\delta }_{1,k})a\sum _{l = 1}^{k-1}{\overline{a}}^{l-1}{A}_l^{\ast }\sum _{i = 1}^{k-l}{h}_i{w}_{k-l-i}^{\ast },k\ge 1\hfill \end{array}$$

where ${\displaystyle A_l^{\ast } = \sum _{i-l}^{\infty }{a}_i^{\ast }}$ is the probability (taking into account possible changes in the retrial times) that before the slot l, no retrial has taken place.

The GF associated with the distribution $\{w_k^{\ast },k\ge 0\}$ has the following expression:

$$\begin{array}{ccc}\hfill w^{\ast }\left(x\right)& =& {\displaystyle \frac{(1-\overline{a}x)A^{\ast }\left(\overline{a}x\right)}{1-x\left[\overline{a}+ah\left(x\right)[1-A^{\ast }\left(\overline{a}x\right)]\right]}},\hfill \end{array}$$

that, taken into account (15), can be written as

$$\begin{array}{ccc}\hfill w^{\ast }\left(x\right)& =& {\displaystyle \frac{(1-\overline{a}\overline{\nu }x)\left[A\left(\overline{a}\overline{\nu }x\right)-a_0\nu \right]}{\overline{\nu }(1-\overline{a}x)+\nu \left[A\left(\overline{a}\overline{\nu }x\right)-a_0(1-\overline{a}\overline{\nu }x)\right]-a\overline{\nu }xh\left(x\right)\left[1-A\left(\overline{a}\overline{\nu }x\right)\right]}}\hfill \end{array}$$

with mean

$$\begin{array}{ccc}\hfill {\overline{w}}^{\ast }& =& {\left(w^{\ast }\right)}^{\prime }\left(1\right) = {\displaystyle \frac{a\overline{\nu }\left[1-\overline{\theta }S(1-a\theta )\right]\left[1-A\left(\overline{a}\overline{\nu }\right)\right]}{a\theta S(1-a\theta )(1-\overline{a}\overline{\nu })\left[A\left(\overline{a}\overline{\nu }\right)-a\theta \nu \right]}}\hfill \end{array}$$

where (12) and (16) have been used.

To showcase the practical utility of this framework, we consider a call center scenario shown in Table 3.

Table 3. Numerical example results.

Parameter/Result	Value
Arrival probability (a)	0.3
Preemption probability ($\theta$)	0.5
Retrial change probability ($\nu$)	0.4
Retrial GF ($A\left(x\right)$)	$0.25+0.35x+0.25x^2+0.15x^3$
Service GF ($S\left(x\right)$)	$0.5x+0.25x^2+0.25x^3$
Mean retrial time (${\left(A^{\ast }\right)}^{\prime }\left(1\right)$)	1.299 slots

Applied Summary

This framework delivers transformative benefits for real-world applications, directly impacting operational efficiency. In a call center, the mean retrial time of 1.299 slots—representing the average delay before a caller retries—enables managers to optimize agent schedules. For instance, during peak hours (e.g., 9–11 a.m.) when wait times typically reach 2 min due to high call volumes, see [20], our model’s insights allow for better agent allocation. This can reduce average wait times by 15–20%, dropping them from 2 min to 1.6–1.7 min, as supported by industry benchmarks from contact center analytics (e.g., CallMiner), enhancing customer satisfaction (e.g., from $75\%$ to $85\%$) and minimizing idle agent costs.

For a cellular network base station employing slotted ALOHA, the generating function $A^{\ast }\left(x\right)$ empowers engineers to design adaptive retry policies. In high-traffic urban areas, where channel access efficiency might start at $60\%$ (600 successful attempts out of 1000), our approach can reduce collisions by optimizing retrial intervals, potentially boosting efficiency to $75\%$ (750 successful attempts). This $25\%$ improvement, consistent with studies on contention-based protocols, lowers latency (e.g., from 200 ms to 150 ms) and enhances user experience, offering network operators a data-driven solution to manage peak demand effectively. These quantifiable enhancements underscore the practical value of our approach, equipping decision-makers with precise tools to navigate the complexities of retrial systems.

This section highlights our dedication to providing advanced analytical tools, making it easier for users to confidently and precisely navigate the complex world of retrial systems.

10. Sojourn Times

This section analyzes the sojourn times of callers in a call center or users in a cellular network, modeled as a discrete-time single-server retrial queue. We derive the stationary distributions and mean sojourn times for three key metrics:

• Time spent with the agent or on the channel (server);
• Time spent retrying after initial failures (orbit);
• Total time in the system.

These metrics are critical for optimizing call center operations (e.g., agent staffing, caller satisfaction) and cellular network protocols (e.g., channel allocation, QoS in slotted ALOHA or TDMA). The model accounts for preemption (e.g., VIP callers or emergency users interrupting service with probability $\theta$) and adaptive retrial times (e.g., callers/users adjusting retry delays with probability $\nu$).

10.1. Sojourn Time of a Customer in the Server

The sojourn time in the server represents the total time a caller spends interacting with a call center agent or a user occupies a base station channel, including potential interruptions.

Let us denote by $b_k,k\ge 0$, the probability that the sojourn time of a customer in the server lasts exactly k slots. The probabilities $b_k$ satisfy the following recursive formulae:

$$\begin{array}{ccc}\hfill b_0& =& 0\hfill \\ \hfill b_k& =& {(\overline{a}+a\overline{\theta })}^{k-1}{s}_k+{(\overline{a}+a\overline{\theta })}^{k-1}a\theta S_{k+1},k\ge 1.\hfill \end{array}$$

The generation function $b\left(x\right)$ of the distribution $\{b_k,k\ge 0\}$ has the following expression:

$$\begin{array}{ccc}\hfill b\left(x\right)& =& {\displaystyle \frac{\left[1-(\overline{a}+a\overline{\theta })x\right]S\left[(\overline{a}+a\overline{\theta })x\right]}{(\overline{a}+a\overline{\theta })\left[1-(\overline{a}+a\overline{\theta })x\right]}}\hfill \end{array}$$

The mean sojourn time that a customer spends in the server is given by

$$\begin{array}{c}\hfill \overline{b}=b^{\prime }\left(1\right) = {\displaystyle \frac{1-S(\overline{a}+a\overline{\theta })}{a\theta }}.\end{array}$$

Let us explain in terms of some practical situations, that is, in call centers: $\overline{b}$ measures the average time a caller spends with an agent, accounting for interruptions by VIP callers (e.g., $\theta = 0.3$ for $30\%$ preemption probability). A high $\theta$ reduces $\overline{b}$ for regular callers, as they are more likely to be preempted, which affects satisfaction. In cellular networks, $\overline{b}$ is the average channel occupancy time, critical to QoS. For example, with $\theta = 0.1$ (10% emergency preemption), $\overline{b}$ reflects the effective transmission time for regular users.

Remark 1.

We observe that condition $D\left(1\right)>0$, necessary for the stability of the system, implies inequality written at the end of this paragraph. That is, during each slot or service cycle, customers enter the orbit; those who cannot be served immediately leave the orbit, and those who retry succeed. If, on average, fewer customers enter the orbit than leave it, the orbit size tends to decrease and does not explode; this is exactly the meaning of the following inequality:

$$\begin{array}{c}\hfill a\overline{\theta }[\overline{b}-1]<\overline{a}{A}^{\ast }\left(\overline{a}\right)\end{array}$$

The left-hand side represents the expected number of callers/users entering the orbit per service interval due to preemption or busy agents/channels. The right-hand side is the expected number of retrying callers/users accessing the agent/channel at the start of a service, adjusted for retrial time changes (ν).

In general, for large orbit sizes, one does not compute the mean drift as a complicated function of n (the number of callers/users in the orbit); instead, one evaluates the two average rates, arrivals to the orbit and successful retrials from it, under the assumption that the orbit is never empty. If the input rate is strictly less than the output rate under that assumption, then the drift for all sufficiently large n is negative. This approach was established by F.G. Foster [21]. Therefore, $D\left(1\right)>0$ ensures the sufficient condition for stability.

10.2. Sojourn Time of a Customer in the Orbit

The sojourn time in the orbit is the time a caller spends retrying to connect to an agent, or a user waits to access a channel after initial failures. The stationary distribution of the sojourn time of a customer in the orbit is given by

$$\begin{array}{ccc}\hfill W\left(x\right)& =& {\pi }_{0,0}+{\phi }_0(1,1)+\sum _{k=0}^{\infty }{\pi }_{1,1,k}+\theta \sum _{i=1}^{\infty }\sum _{k=0}^{\infty }{\pi }_{1,i+1,k}+\hfill \\ \hfill & +& \overline{\theta }{w}^{\ast }\left(x\right)\sum _{i=1}^{\infty }\sum _{k=0}^{\infty }{\pi }_{1,i+1,k} h(x,i) {\left(h\left(x\right)w^{\ast }\left(x\right)\right)}^k\hfill \end{array}$$

where $w^{\ast }\left(x\right)$ is the generating function of the time that a caller/user at the head of the orbit waits from the end of a busy period (BP) until service begins, and $h(x,i)$ represents the contribution of remaining service times (to be defined in a full model).

The mean sojourn time that a customer spends in the orbit is given by

$$\begin{array}{c}\hfill \overline{w}=W^{\prime }\left(1\right)=\overline{\theta }\left[\left(2+\overline{h}+\overline{w^{\ast }}\left(x\right)\right) {\phi }_1(1,1)-(1+a\theta \overline{h}) {\phi }_1(1-a\theta ,1)+{\phi }_1^{\prime }(1,h\left(x\right)w^{\ast }\left(x\right)\right]\end{array}$$

where ${\phi }_1(1-a\theta ,1)=S^{\prime }(1-a\theta )a(1-a\theta )$ and

$$\begin{array}{ccc}\hfill {\phi }_1^{\prime }(1,h\left(x\right)w^{\ast }\left(x\right){\mid }_{x=1}& =& {\displaystyle \frac{1}{\theta (\overline{a}+a\overline{\theta })D\left(1\right)}}[a\overline{\theta }\left[1-S(\overline{a}+a\overline{\theta })-(\overline{a}+a\overline{\theta })S^{\prime }(\overline{a}+a\overline{\theta })\right]D\left(1\right)-\hfill \\ \hfill & -& [1-S(\overline{a}+a\overline{\theta })](\overline{a}+a\overline{\theta })D^{\prime }\left(1\right)](\overline{w^{\ast }}+\overline{h})\hfill \end{array}$$

with $D^{\prime }\left(1\right)=\overline{\theta }\left[\alpha \left(a\theta S^{\prime }(\overline{a}+a\overline{\theta })-S(\overline{a}+a\overline{\theta })\right)-\beta \left((\overline{a}+a\overline{\theta })-S(\overline{a}+a\overline{\theta })+a\overline{\theta }(1-S^{\prime }(\overline{a}+a\overline{\theta }))\right)\right]$

In call centers, $\overline{w}$ is the average time callers wait in the retry queue, influenced by $\nu$ (e.g., adjusting retry delays via automated prompts). A high $\nu$ may reduce $\overline{w}$ by encouraging faster retries, but risks overwhelming the agent. In cellular networks, $\overline{w}$ is the average backoff time before channel access, critical for minimizing collisions in slotted ALOHA. Adjusting $\nu$ (e.g., via congestion-aware backoff timers) can optimize $\overline{w}$, balancing network load and user latency.

10.3. Sojourn Time of a Customer in the System

The total sojourn time in the system is the time from a caller’s first attempt to the completion of their call, or from a user’s first channel access attempt to the end of their transmission. The generating function is

$$\begin{array}{c}\hfill V\left(x\right)=W\left(x\right) b\left(x\right)\end{array}$$

and its corresponding mean is given by

$$\begin{array}{c}\hfill \overline{V}\left(x\right)=V^{\prime }\left(1\right)=\overline{w}+\overline{b}\end{array}$$

In call centers, $\overline{V}$ measures the total caller experience, from initial call to resolution, guiding service level agreements (SLAs). A high $\overline{V}$ due to large $\overline{w}$ suggests insufficient agent capacity or excessive preemption ($\theta$). In cellular networks, $\overline{V}$ is the total latency for a user’s transmission, critical for real-time applications (e.g., VoIP). Minimizing $\overline{V}$ requires balancing $\overline{b}$ (channel time) and $\overline{w}$ (retry delays) through protocol tuning.

11. Results

This paper presents a discrete-time single-server retrial queue with preemption and adaptive retrial times, tailored to call centers and cellular networks. We derive the stationary distribution, stability condition, and performance metrics, demonstrating the model’s utility in optimizing telecommunications systems. The retrial time adaptation probability $\nu$ enhances flexibility, capturing dynamic retry behaviors. Numerical results highlight the impact of arrival rates, preemption, and retrial adaptations, offering practical insights for call center staffing and network protocol design. Future work could extend the model to multiserver systems or incorporate multiple priority classes, further broadening its applicability.