Stochastic Approximations and Monotonicity of a Single Server Feedback Retrial Queue

Abstract

This paper focuses on stochastic comparison of the Markov chains to derive some qualitative approximations for an 𝑀/𝐺/1 retrial queue with a Bernoulli feedback. The main objective is to use stochastic ordering techniques to establish various monotonicity results with respect to arrival rates, service time distributions, and retrial parameters.

1. Introduction

Retrial queueing systems are described by the feature that the arriving customers (or calls) who find the server busy join the orbit to try again for their requests in a random order and at random time intervals. Retrial queues are widely and successfully used as mathematical models of several computer systems and telecommunication networks. For excellent and recent bibliographies on retrial queues, the reader is referred to [1–3].

Most of the queueing systems with repeated attempts assume that each customer in the retrial group seeks service independently of each other after a random time exponentially distributed with rate 𝜃 so that the probability of repeated attempt during the interval (𝑡,𝑡+Δ𝑡) given that there were 𝑛 customers in orbit at time 𝑡 is 𝑛𝜃Δ𝑡+∘(Δ𝑡). This discipline for access to the server from the retrial group is called classical retrial policy [4, 5].

Several papers on retrial queues have analyzed the systems without customer feedback. A more practical retrial queue with the Bernoulli feedback of the customers occurs in many real world situations: for instance, in communication networks where data transmissions need to be guaranteed to be error free within some specified probability, feedback schemes are used to request retransmission of packets that are lost or received in a corrupted form.

Because of complexity of retrial queueing models, analytic results are generally difficult to obtain. In contrast, there are a great number of numerical and approximation methods which are of practical importance. One important approach is monotonicity which allow to establish some stochastic bounds helpful in understanding complicated models by more simpler models for which an evaluation can be made using the stochastic comparison method based on the general theory of stochastic orderings [6].

Stochastic orders represent an important tool for many problems in probability and statistics. They lead to powerful approximation methods and bounds in situations where realistic stochastic models are too complex for rigorous treatment. They are also helpful in situations where fundamental model distributions are only known partially. Further details and applications about these stochastic orders may be found in [6–8].

There exists a flourishing literature on stochastic comparison methods and monotonicity of queues. Oukid and Aissani [9] obtain lower bound and new upper bound for the mean busy period of 𝐺𝐼/𝐺𝐼/1 queue with breakdowns and FIFO discipline. Boualem et al. [10] investigate some monotonicity properties of an 𝑀/𝐺/1 queue with constant retrial policy in which the server operates under a general exhaustive service and multiple vacation policy relative to strong stochastic ordering and convex ordering. These results imply in particular simple insensitive bounds for the stationary queue length distribution. More recently, Taleb and Aissani [11] investigate some monotonicity properties of an unreliable 𝑀/𝐺/1 retrial queue relative to strong stochastic ordering and increasing convex ordering.

In this work, we use the tools of the qualitative analysis to investigate various monotonicity properties for an 𝑀/𝐺/1 retrial queue with classical retrial policy and Bernoulli feedback relative to strong stochastic ordering, increasing convex ordering and the Laplace ordering. Instead of studying a performance measure in a quantitative fashion, this approach attempts to reveal the relationship between the performance measures and the parameters of the system.

The rest of the paper is organized as follows. In Section 2, we describe the mathematical model in detail and derive the generating function of the stationary distribution. In Section 3, we present some useful lemmas that will be used in what follows. Section 4 focusses on stochastic monotonicity of the transition operator of the embedded Markov chain and gives comparability conditions of two transition operators. Stochastic bounds for the stationary number of customers in the system are discussed in Section 5. In Section 6, we obtain approximations for the conditional distribution of the stationary queue given that the server is idle.

2. Description and Analysis of the Queueing System

We consider a single server retrial queue with the Bernoulli feedback at which customers arrive from outside the system according to a Poisson process with rate 𝜆. An arriving customer receives immediate service if the server is idle, otherwise he leaves the service area temporarily to join the retrial group. Any orbiting customer produces a Poisson stream of repeated calls with intensity 𝜃 until the time at which he finds the server idle and starts his service. The service times follow a general probability law with distribution function 𝐵(𝑥) having finite mean 𝛽₁ and Laplace-Stieltjes transform 𝛽(𝑠). After the customer is completely served, he will decide either to join the retrial group again for another service with probability 𝑝 (0<𝑝<1) or to leave the system forever with probability 𝑝(=1−𝑝).

We finally assume that the input flow of primary arrivals, intervals between repeated attempts and service times, are mutually independent.

The state of the system at time 𝑡 can be described by the Markov process 𝑅(𝑡)={𝐶(𝑡),𝑁(𝑡),𝜁(𝑡)}_(𝑡≥0), where 𝐶(𝑡) is the indicator function of the server state: 𝐶(𝑡) is equal to 0 or 1 depending on whether the server is free or busy at time 𝑡 and 𝑁(𝑡) is the number of customers occupying the orbit. If 𝐶(𝑡)=1, then 𝜁(𝑡) corresponds to the elapsed time of the customer being served at time 𝑡.

Note that the stationary distribution of the system state (the stationary joint distribution of the server state and the number of customers in the orbit) was found in [12], using the supplementary variables method. In this section, we are interested in the embedded Markov chain. To this end, we describe the structure of the latter, determine its ergodicity condition, and obtain its stationary distribution.

2.1. Embedded Markov Chain

Let 𝜏_𝑛 be the time of the 𝑛th departure and 𝐷_𝑛 the number of customers in the orbit just after the time 𝜏_𝑛, then 𝐶(𝜏⁺_𝑛)=0 and 𝑁(𝜏⁺_𝑛)=𝐷_𝑛, ∀𝑛≥1. We have the following fundamental recursive equation: 𝐷_𝑛+1=𝐷_𝑛+𝑣_𝑛+1−𝛿_𝐷𝑛+1+𝜂, (2.1) where (i) 𝑣_𝑛+1 is the number of primary customers arriving at the system during the service time which ends at 𝜏_𝑛+1. It does not depend on events which have occurred before the beginning of the (𝑛+1)st service. Its distribution is given by: 𝑘_𝑗=𝑃𝑣_𝑛+1=𝑗=^∞₀(𝜆𝑥)^𝑗𝑒𝑗!^−𝜆𝑥𝑑𝐵(𝑥),𝑗≥0, (2.2) with generating function 𝐾(𝑧)=∑_𝑗≥0𝑘_𝑗𝑧^𝑗=𝛽(𝜆(1−𝑧)),

(ii) the Bernoulli random variable 𝛿_𝐷𝑛+1 is equal to 1 or 0 depending on whether the customer who leaves the service area at time 𝜏_𝑛+1 proceeds from the orbit or otherwise. Its conditional distribution is given by 𝑃𝛿_𝐷𝑛+1=1/𝐷_𝑛=𝑙=𝑙𝜃𝛿𝜆+𝑙𝜃,𝑃_𝐷𝑛+1=0/𝐷_𝑛=𝜆=𝑙,𝜆+𝑙𝜃 (2.3)

(iii) the random variable 𝜂 is 0 or 1 depending on whether the served customer leaves the system or goes to orbit. We have also that 𝑃[𝜂=1]=𝑝 and 𝑃[𝜂=0]=𝑝(=1−𝑝).

The sequence {𝐷_𝑛,𝑛≥1} forms an embedded Markov chain with transition probability matrix 𝐏=(𝑝_𝑖𝑗)_{𝑖,𝑗≥0}, where 𝑝_𝑖𝑗=𝑃(𝐷_𝑛+1=𝑗/𝐷_𝑛=𝑖), defined by𝑝_𝑖𝑗=𝜆𝑝𝑘𝜆+𝑖𝜃^𝑗−𝑖+𝑖𝜃𝑝𝑘𝜆+𝑖𝜃^{𝑗−𝑖+1}+𝜆𝑝𝑘𝜆+𝑖𝜃^{𝑗−𝑖−1}+𝑖𝜃𝑝𝑘𝜆+𝑖𝜃^𝑗−𝑖. (2.4) Note that 𝑝_𝑖𝑗≠0 only for 𝑖=0,1,…,𝑗+1.

Theorem 2.1. The embedded Markov chain {𝐷_𝑛,𝑛≥1} is ergodic if and only if 𝜌=𝜆𝛽₁+𝑝<1.

Proof. It is not difficult to see that {𝐷_𝑛,𝑛≥1} is irreducible and aperiodic. To find a sufficient condition, we use Foster's criterion which consists to show the existence of a nonnegative function 𝑓(𝑠), 𝑠∈𝒮, and 𝜖>0 such that the mean drift 𝑥_𝑠=𝐸[𝑓(𝐷_𝑛+1)−𝑓(𝐷_𝑛)/𝐷_𝑛=𝑠] is finite for all 𝑠∈𝒮 and 𝑥_𝑠≤−𝜖 for all 𝑠∈𝒮 except perhaps a finite number. In our case, we consider the function 𝑓(𝑠)=𝑠 for all 𝑠∈𝒮. Then, the mean drift is given by 𝑥_𝑠=𝐸𝑓𝐷_𝑛+1−𝑓𝐷_𝑛/𝐷_𝑛=𝑠=𝐸𝑣_𝑛+1−𝛿_𝐷𝑛+1+𝜂/𝐷_𝑛=𝑠=𝜆𝛽₁−𝑠𝜃𝜆+𝑠𝜃+𝑝. (2.5) Let 𝑥=lim_𝑠→∞𝑥_𝑠. Then 𝑥=𝜆𝛽₁−1+𝑝<0. Therefore, the sufficient condition is 𝜆𝛽₁+𝑃<1.
To prove that the previous condition is also a necessary condition for ergodicity of our embedded Markov chain, we apply Kaplan's condition: 𝑥_𝑖<∞, for all 𝑖≥0, and there is an 𝑖₀ such that 𝑥_𝑖≥0, for 𝑖≥𝑖₀. In our case, this condition is verified because 𝑝_𝑖𝑗=0 for 𝑗<𝑖−1 and 𝑖>0 (see (2.4)).

2.2. Generating Function of the Stationary Distribution

Now, under the condition 𝜌<1, we find the stationary distribution 𝜋_𝑚=lim_𝑛→∞𝑃(𝐷_𝑛=𝑚). Using (2.4), one can obtain Kolmogorov equations for the distribution 𝜋_𝑚𝜋_𝑚=^𝑚_𝑛=0𝜆𝑝𝜋𝜆+𝑛𝜃^𝑛𝑘_𝑚−𝑛+^𝑚+1_𝑛=1𝑛𝜃𝑝𝜋𝜆+𝑛𝜃^𝑛𝑘_{𝑚−𝑛+1}+^𝑚−1_𝑛=0𝜆𝑝𝜋𝜆+𝑛𝜃^𝑛𝑘_{𝑚−𝑛−1}+^𝑚_𝑛=0𝑛𝜃𝑝𝜋𝜆+𝑛𝜃^𝑛𝑘_𝑚−𝑛. (2.6) Because of presence of convolutions, these equations can be transformed with the help of the generating functions 𝜋(𝑧)=∑_𝑚≥0𝜋_𝑚𝑧^𝑚and𝐿(𝑧)=∑_𝑚≥0(𝜋_𝑚/𝜆+𝑚𝜃)𝑧^𝑚 to𝜋(𝑧)=𝛽(𝜆−𝜆𝑧)𝜆𝑝𝐿(𝑧)+𝜃𝑧𝑝𝐿^′(𝑧)+𝜃𝑝𝐿^′(𝑧)(𝑧=𝛽()+𝜆𝑝𝑧𝐿)𝜆−𝜆𝑧𝑝+𝑝𝑧𝜆L(𝑧)+𝜃𝐿^′(𝑧). (2.7) Since𝜋(𝑧)=𝜆𝜋_𝑚𝑧𝜆+𝑚𝜃^𝑚+𝑚𝜃𝜋_𝑚𝑧𝜆+𝑚𝜃^𝑚=𝜆𝐿(𝑧)+𝜃𝑧𝐿^′(𝑧), (2.8) from (2.7) and (2.8), we have𝜃𝐿^′(𝑧)𝑝+𝑝𝑧𝛽(𝜆−𝜆𝑧)−𝑧=𝜆𝐿(𝑧)1−𝑝+𝑝𝑧𝛽(𝜆−𝜆𝑧). (2.9) We consider now the function 𝑓(𝑧)=(𝑝+𝑝𝑧)𝛽(𝜆−𝜆𝑧)−𝑧.

It is easy to show that 𝑓(1)=𝛽(0)𝑓−1=1−1=0,_′(𝑧)=−𝜆𝑝+𝑝𝑧𝛽^′()𝛽()𝑓𝜆−𝜆𝑧+𝑝𝜆−𝜆𝑧−1,_′(1)=−𝜆𝛽^′(0)𝛽(0)𝑓+𝑝−1=𝜌−1<0,_′′(𝑧)=𝜆²𝑝+𝑝𝑧𝛽^′′(𝜆−𝜆𝑧)≥0. (2.10) Therefore the function 𝑓(𝑧) is decreasing on the interval [0,1], 𝑧=1 is the only zero there and for 𝑧∈[0,1) the function is positive, that is, (as 𝜌<1) for 𝑧∈[0,1) we have: 𝑧<(𝑝+𝑝𝑧)𝛽(𝜆−𝜆𝑧)≤1.

Besides, 1−𝑝+𝑝𝑧𝛽(𝜆−𝜆𝑧)𝛽()=𝜌𝑝+𝑝𝑧𝜆−𝜆𝑧−𝑧1−𝜌<∞, (2.11) that is, the function 1−(𝑝+𝑝𝑧)𝛽(𝜆−𝜆𝑧)/(𝑝+𝑝𝑧)𝛽(𝜆−𝜆𝑧)−𝑧 can be defined at the point 𝑧=1 as 𝜌/1−𝜌.

This means that for 𝑧∈[0,1] we can rewrite (2.9) as follows:𝐿^′(𝑧)=𝜆𝜃1−𝑝+𝑝𝑧𝛽(𝜆−𝜆𝑧)𝛽()𝐿(𝑧)𝑝+𝑝𝑧𝜆−𝜆𝑧−𝑧. (2.12) The solution of the differential equation (2.12) is given by 𝐿(𝑧)(1)−𝜆=𝐿exp𝜃¹_𝑧1−𝑝+𝑝𝑢𝛽(𝜆−𝜆𝑢)𝑝+𝑝𝑢𝛽(𝜆−𝜆𝑢)−𝑢𝑑𝑢. (2.13) From (2.8), we have 𝜋(𝑧)=𝜆𝑝+𝑝𝑧𝛽(𝜆−𝜆𝑧)(1−𝑧)𝛽()𝐿(1)−𝜆𝑝+𝑝𝑧𝜆−𝜆𝑧−𝑧exp𝜃¹_𝑧1−𝑝+𝑝𝑢𝛽(𝜆−𝜆𝑢)𝑝+𝑝𝑢𝛽(𝜆−𝜆𝑢)−𝑢𝑑𝑢. (2.14) We obtain from the normalization condition 𝜋(1)=1 that 𝐿(1)=(1−𝜌)/𝜆.

Finally we get the following formula for the generating function of the steady state queue size distribution at departure epochs (which is known in the literature as the stochastic decomposition property):𝜋(𝑧)=1−𝜌𝛽(𝜆−𝜆𝑧)(1−𝑧)𝑝+p𝑧𝛽(𝜆−𝜆𝑧)−𝑧−𝜆𝑝+𝑝𝑧exp𝜃¹_𝑧1−𝑝+𝑝𝑢𝛽(𝜆−𝜆𝑢)𝑝+𝑝𝑢𝛽(𝜆−𝜆𝑢)−𝑢𝑑𝑢. (2.15) It is easy to see that the right hand part of expression (2.15) can be decomposed into two factors. The first factor is the generating function for the number of customers in 𝑀/𝐺/1 queueing system with Bernoulli feedback (see [13]); the remaining one is the generating function for the number of customers in the retrial queue with feedback given that the server is idle [12]. One can see that formula (2.15) is cumbersome (it includes integrals of Laplace transform, solutions of functional equations). It is why we use, in the rest of the paper, the general theory of stochastic orderings to investigate the monotonicity properties of the system relative to the strong stochastic ordering, convex ordering, and Laplace ordering.

3. Preliminaries

3.1. Stochastic Orders and Ageing Notions

First, let us recall some stochastic orders and ageing notions which are most pertinent to the main results to be developed in the subsequent section.

Definition 3.1. For two random variables 𝑋 and 𝑌 with densities 𝑓 and 𝑔 and cumulative distribution functions 𝐹 and 𝐺, respectively, let 𝐹=1−𝐹 and 𝐺=1−𝐺 be the survival functions. As the ratios in the statements below are well defined, 𝑋 is said to be smaller than 𝑌 in: (a)stochastic ordering (denoted by 𝑋≤_st𝑌) if and only if 𝐹(𝑥)≤𝐺(𝑥), ∀𝑥≥0, (b)increasing convex ordering (denoted by 𝑋≤_icx𝑌) if and only if ∫_𝑥^+∞𝐹(𝑢)𝑑(𝑢)≤∫_𝑥^+∞𝐺(𝑢)𝑑(𝑢),∀𝑥≥0, (c)Laplace ordering (denoted by 𝑋≤_𝐿𝑌) if and only if ∫₀^+∞exp(−𝑠𝑥)𝑑𝐹(𝑥)≥∫₀^+∞exp(−𝑠𝑥)𝑑𝐺(𝑥),∀𝑠≥0. If the random variables of interest are of discrete type and 𝜔=(𝜔_𝑛)_𝑛≥0, 𝛽=(𝛽_𝑛)_𝑛≥0 are the corresponding distributions, then the definitions above can be given in the following forms: (a)𝜔≤_st𝛽 if and only if 𝜔_𝑚=∑_𝑛≥𝑚𝜔_𝑛≤𝛽_𝑚=∑_𝑛≥𝑚𝛽_𝑛, for all 𝑚, (b)𝜔≤_icx𝛽 if and only if 𝜔_𝑚=∑_𝑛≥𝑚∑_𝑘≥𝑛𝜔_𝑘≤𝛽_𝑚=∑_𝑛≥𝑚∑_𝑘≥𝑛𝛽_𝑘, for all 𝑚, (c)𝜔≤_𝐿𝛽 if and only if ∑_𝑛≥0𝜔_𝑛𝑧^𝑛≥∑_𝑛≥0𝛽_𝑛𝑧^𝑛, for all 𝑧∈[0,1].

For a comprehensive discussion on these stochastic orders see [6–8].

Definition 3.2. Let 𝑋 be a positive random variable with distribution function 𝐹: (a)𝐹 is HNBUE (harmonically new better than used in expectation) if and only if 𝐹≤_icx𝐹^∗, (b)𝐹 is HNWUE (harmonically new worse than used in expectation) if and only if 𝐹≥_icx𝐹^∗, (c)𝐹 is of class ℒ if and only if 𝐹≥_𝑠𝐹^∗, where 𝐹^∗ is the exponential distribution function with the same mean as 𝐹.

3.2. Some Useful Lemmas

Consider two 𝑀/𝐺/1 retrial queues with classical retrial policy and Bernoulli feedback with parameters 𝜆^(𝑖) and 𝐵^(𝑖), 𝑖=1,2. Let 𝑘_𝑗^(𝑖)=∫₀^+∞((𝜆^(𝑖)𝑥)^𝑗/𝑗!)𝑒^{−𝜆(𝑖)𝑥}𝑑𝐵^(𝑖)(𝑥) be the distribution of the number of primary calls which arrive during the service time of a call in the 𝑖th system.

The following lemma turns out to be a useful tool for showing the monotonicity properties of the embedded Markov chain.

Lemma 3.3. If 𝜆⁽¹⁾≤𝜆⁽²⁾ and 𝐵⁽¹⁾≤_𝑠𝐵⁽²⁾, then {𝑘_𝑛⁽¹⁾}≤_𝑠{𝑘_𝑛⁽²⁾}, where ≤_𝑠 is either ≤_st or ≤_icx.

Proof. To prove that {𝑘_𝑛⁽¹⁾}≤_𝑠{𝑘_𝑛⁽²⁾}, we have to establish the usual numerical inequalities: 𝑘_𝑛⁽¹⁾=𝑘_𝑚⁽¹⁾≤𝑘_𝑛⁽²⁾,for≤_𝑠=≤_st−ordering,𝑘_𝑛⁽¹⁾=𝑘_𝑚⁽¹⁾≤𝑘_𝑛⁽²⁾,for≤_𝑠=≤_icx−ordering. (3.1) The rest of the proof is known in the more general setting of a random summation.

The next lemma is key to proving the main result in Section 6.

Lemma 3.4. If 𝜆⁽¹⁾≤𝜆⁽²⁾ and 𝐵⁽¹⁾≤_𝐿𝐵⁽²⁾, then {𝑘_𝑛⁽¹⁾}≤_𝐿{𝑘_𝑛⁽²⁾}.

Proof. We have, 𝐾^(𝑖)(𝑧)=∑_𝑛≥0𝑘_𝑛^(𝑖)𝑧^𝑛=𝛽^(𝑖)(𝜆^(𝑖)(1−𝑧)),𝑖=1,2, where 𝐾⁽¹⁾(𝑧), 𝐾⁽²⁾(𝑧) are the corresponding distributions of the number of new arrivals during a service time.
Let 𝜆⁽¹⁾≤𝜆⁽²⁾, 𝐵⁽¹⁾≤_𝐿𝐵⁽²⁾. To prove that {𝑘_𝑛⁽¹⁾}≤_𝐿{𝑘_𝑛⁽²⁾}, we have to establish that 𝛽⁽¹⁾𝜆⁽¹⁾(1−𝑧)≥𝛽⁽²⁾𝜆⁽²⁾(1−𝑧). (3.2) The inequality 𝐵⁽¹⁾≤_𝐿𝐵⁽²⁾ means that 𝛽⁽¹⁾(𝑠)≥𝛽⁽²⁾(𝑠) for all 𝑠≥0.
In particular, for 𝑠=𝜆⁽¹⁾(1−𝑧) we have 𝛽⁽¹⁾𝜆⁽¹⁾(1−𝑧)≥𝛽⁽²⁾𝜆⁽¹⁾(1−𝑧). (3.3) Since any Laplace transform is a decreasing function, 𝜆⁽¹⁾≤𝜆⁽²⁾ implies that 𝛽⁽²⁾𝜆⁽¹⁾(1−𝑧)≥𝛽⁽²⁾𝜆⁽²⁾(1−𝑧). (3.4) By transitivity, (3.3) and (3.4) give (3.2).

4. Stochastic Monotonicity of Transition Operator

Let 𝐐 be the transition operator of an embedded Markov chain, which associates to every distribution 𝜔={𝑝_𝑖}_𝑖≥0, a distribution 𝐐_𝜔={𝑞_𝑗}_𝑗≥0 such that 𝑞_𝑗=∑_𝑖𝑝_𝑖𝑝_𝑖𝑗.

Corollary 4.1 (see [6]). The operator 𝐐 is monotone with respect to ≤_st if and only if 𝑝_𝑛𝑚−𝑝_𝑛−1𝑚≥0, and is monotone with respect to ≤_icx if and only if 𝑝_𝑛−1𝑚+𝑝_𝑛+1𝑚−2𝑝_𝑛𝑚≥0,∀𝑛,𝑚. Here, 𝑝_𝑛𝑚=∑^∞_𝑙=𝑚𝑝_𝑛𝑙 and 𝑝_𝑛𝑚=∑^∞_𝑙=𝑚𝑝_𝑛𝑙.

Theorem 4.2. The transition operator 𝐐 of the embedded chain {𝐷_𝑛,𝑛≥1} is monotone with respect to the orders ≤_st and ≤_icx.

Proof. We have 𝑝_𝑛𝑚=𝑝_𝑛𝑘=𝑘_𝑚−𝑛+𝜆𝑝𝑘𝜆+𝑛𝜃^{𝑚−𝑛−1}−𝑛𝜃𝑝𝑘𝜆+𝑛𝜃^𝑚−𝑛,𝑝_𝑛𝑚=𝑟_𝑛𝑘=𝑘_𝑚−𝑛+𝜆𝑝𝜆+𝑛𝜃𝑘_{𝑚−𝑛−1}−𝑛𝜃𝑝𝜆+𝑛𝜃𝑘_𝑚−𝑛. (4.1) Thus 𝑝_𝑛𝑚−𝑝_𝑛−1𝑚=𝜆²𝑝+(𝑛−1)𝜆𝜃+𝑛(𝑛−1)𝜃²𝑝(()𝜃)𝑘𝜆+𝑛𝜃)(𝜆+𝑛−1^𝑚−𝑛+(𝑛−1)𝜃𝑝()𝜃𝑘𝜆+𝑛−1^{𝑚−𝑛+1}+𝜆𝑝𝑘𝜆+𝑛𝜃^{𝑚−𝑛−1}≥0,𝑝_𝑛−1𝑚+𝑝_𝑛+1𝑚−2𝑝_𝑛𝑚=𝜆𝑝()𝜃𝑘𝜆+𝑛+1^{𝑚−𝑛−2}+(𝑛−1)𝜃𝑝()𝜃𝑘𝜆+𝑛−1^𝑚−𝑛+𝜆²𝑝+𝑛−𝑝𝜆𝜃+𝑛(𝑛+1)𝜃²𝑝(()𝜃)𝑘𝜆+𝑛𝜃)(𝜆+𝑛+1^{𝑚−𝑛−1}+2𝜃²(𝜆+𝑛𝜃)(𝜆+(𝑛−1)𝜃)(𝜆+(𝑛+1)𝜃)𝑘_𝑚−𝑛≥0. (4.2) Based on Corollary 4.1 we obtain the stated result.

In Theorem 4.3, we give comparability conditions of two transition operators. Consider two 𝑀/𝐺/1 retrial queues with classical retrial policy and feedback with parameters 𝜆⁽¹⁾,𝜃⁽¹⁾,𝑝⁽¹⁾,𝐵⁽¹⁾, and 𝜆⁽²⁾,𝜃⁽²⁾,𝑝⁽²⁾,𝐵⁽²⁾, respectively. Let 𝐐¹, 𝐐² be the transition operators of the corresponding embedded Markov chains.

Theorem 4.3. If 𝜆⁽¹⁾≤𝜆⁽²⁾,𝜃⁽¹⁾≥𝜃⁽²⁾,𝑝⁽¹⁾≤𝑝⁽²⁾,𝐵⁽¹⁾≤_𝑠𝐵⁽²⁾, where ≤_𝑠 is either ≤_st or ≤_icx, then 𝐐¹≤_𝑠𝐐², that is, for any distribution 𝜔, one has 𝐐¹𝜔≤_𝑠𝐐²𝜔.

Proof. From Stoyan [6], we wish to establish that 𝑝⁽¹⁾_𝑛𝑚≤𝑝⁽²⁾_𝑛𝑚,∀𝑛,𝑚,for≤_𝑠=≤_st−ordering,𝑝⁽¹⁾_𝑛𝑚≤𝑝⁽²⁾_𝑛𝑚,∀𝑛,𝑚,for≤_𝑠=≤_icx−ordering. (4.3)(4.4) To prove inequality (4.3), we have (for 𝑖=1,2) 𝑝^(𝑖)_𝑛𝑚=𝑘^(𝑖)_𝑚−𝑛+𝜆^(𝑖)𝑝^(𝑖)𝜆^(𝑖)+𝑛𝜃^(𝑖)𝑘^(𝑖)_{𝑚−𝑛−1}−𝑛𝜃^(𝑖)𝑝^(𝑖)𝜆^(𝑖)+𝑛𝜃^(𝑖)𝑘^(𝑖)_𝑚−𝑛. (4.5) By hypothesis, we have that 𝜆⁽¹⁾≤𝜆⁽²⁾,𝜃⁽¹⁾≥𝜃⁽²⁾⟹𝜆⁽¹⁾𝜃⁽¹⁾≤𝜆⁽²⁾𝜃⁽²⁾𝜃,or₍₁₎𝜆⁽¹⁾≥𝜃⁽²⁾𝜆⁽²⁾. (4.6) Since the function 𝐺(𝑥)=𝑥/(𝑥+𝑛) is increasing, we have 𝐺𝜆⁽¹⁾𝜃⁽¹⁾=𝜆⁽¹⁾𝜆⁽¹⁾+𝑛𝜃⁽¹⁾𝜆≤𝐺⁽²⁾𝜃⁽²⁾=𝜆⁽²⁾𝜆⁽²⁾+𝑛𝜃⁽²⁾. (4.7) Moreover, 𝑝⁽¹⁾≤𝑝⁽²⁾. Then 𝜆⁽¹⁾𝜆⁽¹⁾+𝑛𝜃⁽¹⁾𝑝⁽¹⁾≤𝜆⁽²⁾𝜆⁽²⁾+𝑛𝜃⁽²⁾𝑝⁽²⁾. (4.8) Similarly, the function 𝐻(𝑥)=𝑥/(1+𝑥) is increasing, we have 𝐻𝑛𝜃⁽¹⁾𝜆⁽¹⁾=𝑛𝜃⁽¹⁾𝜆⁽¹⁾+𝑛𝜃⁽¹⁾≥𝐻𝑛𝜃⁽²⁾𝜆⁽²⁾=𝑛𝜃⁽²⁾𝜆⁽²⁾+𝑛𝜃⁽²⁾. (4.9) Besides, 𝑝⁽¹⁾≤𝑝⁽²⁾ implies that 𝑝⁽¹⁾≥𝑝⁽²⁾. Hence −𝑛𝜃⁽¹⁾𝜆⁽¹⁾+𝑛𝜃⁽¹⁾𝑝⁽¹⁾≤−𝑛𝜃⁽²⁾𝜆⁽²⁾+𝑛𝜃⁽²⁾𝑝⁽²⁾. (4.10) Using inequalities (4.8)–(4.10) and Lemma 3.3 (for ≤_𝑠=≤_st-ordering) we get 𝑝⁽¹⁾_𝑛𝑚=𝑘⁽¹⁾_𝑚−𝑛+𝜆⁽¹⁾𝑝⁽¹⁾𝜆⁽¹⁾+𝑛𝜃⁽¹⁾𝑘⁽¹⁾_{𝑚−𝑛−1}−𝑛𝜃⁽¹⁾𝑝⁽¹⁾𝜆⁽¹⁾+𝑛𝜃⁽¹⁾𝑘⁽¹⁾_𝑚−𝑛≤𝑘⁽¹⁾_𝑚−𝑛+𝜆⁽²⁾𝑝⁽²⁾𝜆⁽²⁾+𝑛𝜃⁽²⁾𝑘⁽¹⁾_{𝑚−𝑛−1}−𝑛𝜃⁽²⁾𝑝⁽²⁾𝜆⁽²⁾+𝑛𝜃⁽²⁾𝑘⁽¹⁾_𝑚−𝑛=𝜆⁽²⁾𝑝⁽²⁾𝜆⁽²⁾+𝑛𝜃⁽²⁾𝑘⁽¹⁾_𝑚−𝑛+𝑛𝜃⁽²⁾𝑝⁽²⁾𝜆⁽²⁾+𝑛𝜃⁽²⁾𝑘⁽¹⁾_{𝑚−𝑛+1}+𝜆⁽²⁾𝑝⁽²⁾𝜆⁽²⁾+𝑛𝜃⁽²⁾𝑘⁽¹⁾_{𝑚−𝑛−1}+𝑛𝜃⁽²⁾𝑝⁽²⁾𝜆⁽²⁾+𝑛𝜃⁽²⁾𝑘⁽¹⁾_𝑚−𝑛≤𝑝⁽²⁾_𝑛𝑚. (4.11) Following the technique above and using Lemma 3.3 (for ≤_𝑠=≤_icx-ordering), we establish inequality (4.4).

5. Stochastic Bounds for the Stationary Distribution

Consider two 𝑀/𝐺/1 retrial queues with classical retrial policy and feedback with parameters 𝜆⁽¹⁾,𝜃⁽¹⁾,𝑝⁽¹⁾,𝐵⁽¹⁾ and 𝜆⁽²⁾,𝜃⁽²⁾,𝑝⁽²⁾,𝐵⁽²⁾, respectively, and let 𝜋_𝑛⁽¹⁾, 𝜋_𝑛⁽²⁾ be the corresponding stationary distributions of the number of customers in the system.

Theorem 5.1. If 𝜆⁽¹⁾≤𝜆⁽²⁾,𝜃⁽¹⁾≥𝜃⁽²⁾,𝑝⁽¹⁾≤𝑝⁽²⁾,𝐵⁽¹⁾≤_𝑠𝐵⁽²⁾, then {𝜋_𝑛⁽¹⁾}≤_𝑠{𝜋_𝑛⁽²⁾}, where ≤_𝑠 is one of the symbols ≤_st or ≤_icx.

Proof. Using Theorems 4.2 and 4.3 which state, respectively, that 𝐐^𝑖 are monotone with respect to the order ≤_𝑠 and 𝐐¹≤_𝑠𝐐², we have by induction 𝐐^1,𝑛𝜔≤_𝑠𝐐^2,𝑛𝜔 for any distribution 𝜔, where 𝐐^𝑖,𝑛=𝐐^𝑖(𝐐^{𝑖,𝑛−1}𝜔). Taking the limit, we obtain the stated result.

Based on Theorem 5.1 we can establish insensitive stochastic bounds for the generating function of the stationary distribution of the embedded Markov chain defined in (2.15).

Theorem 5.2. For any 𝑀/𝐺/1 retrial queue with classical retrial policy and Bernoulli feedback the distribution 𝜋_𝑛 is greater relative to the increasing convex ordering than the distribution with the generating function 𝜋^∗(𝑧)=1−𝜌𝑒^{𝜆𝛽1(𝑧−1)}(1−𝑧)𝑝+𝑝𝑧𝑒^{𝜆𝛽1(𝑧−1)}−𝑧𝜆𝑝+𝑝𝑧exp𝜃^𝑧₁1−𝑝+𝑝𝑢𝑒^{𝜆𝛽1(𝑢−1)}𝑝+𝑝𝑢𝑒^{𝜆𝛽1(𝑢−1)}.−𝑢𝑑𝑢 (5.1)

Proof. Consider an auxiliary 𝑀/𝐷/1 retrial queue with classical retrial policy and feedback having the same arrival rate 𝜆, retrial rate 𝜃, mean service time 𝛽₁, and probability 𝑝, as those of the 𝑀/𝐺/1 retrial queue with classical retrial policy and Bernoulli feedback. The service times follow a deterministic low with distribution function: 𝐵^⋆(𝑥)=0,if𝑥≤𝛽₁,1,if𝑥>𝛽₁. (5.2) From Stoyan [6], it is known that 𝐵^⋆(𝑥)≤_icx𝐵(𝑥). Therefore, the required result follows from Theorem 5.1.

Theorem 5.3. If in the 𝑀/𝐺/1 retrial queue with classical retrial policy and feedback the service time distribution 𝐵(𝑥) is HNBUE (or HNWUE), then {𝜋_𝑛}≤_icx{𝜋^∗_𝑛} (or {𝜋^∗_𝑛}≤_icx{𝜋_𝑛}), where {𝜋^∗_𝑛} is the stationary distribution of the number of customers in the 𝑀/𝑀/1 retrial queue with classical retrial policy and Bernoulli feedback with the same parameters as those of the 𝑀/𝐺/1 retrial queue with classical retrial policy and Bernoulli feedback.

Proof. Consider an auxiliary 𝑀/𝑀/1 retrial queue with classical retrial policy and Bernoulli feedback with the same arrival rate 𝜆, probability 𝑝, retrial rate 𝜃, and mean service time 𝛽₁ as in the 𝑀/𝐺/1 retrial queue with classical retrial policy and Bernoulli feedback, but with exponentially distributed service time 𝐵^∗(𝑥)=1−exp(−(𝑥/𝛽₁)). If 𝐵(𝑥) is 𝐻𝑁𝐵𝑈𝐸, then 𝐵(𝑥)≤_icx𝐵^∗(𝑥) (if 𝐵(𝑥) is 𝐻𝑁𝑊𝑈𝐸, then 𝐵^∗(𝑥)≤_icx𝐵(𝑥)). Therefore, by using Theorem 5.1, we deduce the statement of this theorem.

6. Stochastic Approximations for the Conditional Distribution

We consider the conditional distribution 𝜑_𝑛 of the stationary queue given that the server is idle. This distribution has also appeared in the stochastic decomposition law for the stationary queue length, see equation (2.15). As we saw its generating function 𝜙(𝑧)=∑_𝑛≥0𝜑_𝑛𝑧^𝑛 was given by 𝜙(𝑧)=−𝜆𝑝+𝑝𝑧exp𝜃¹_𝑧1−𝑝+𝑝𝑢𝛽(𝜆−𝜆𝑢)𝑝+𝑝𝑢𝛽(𝜆−𝜆𝑢)−𝑢𝑑𝑢. (6.1)

Theorem 6.1. Suppose we have two 𝑀/𝐺/1 retrial queues with classical retrial policy and Bernoulli feedback with parameters 𝜆⁽¹⁾,𝜃⁽¹⁾,𝑝⁽¹⁾,𝐵⁽¹⁾ and 𝜆⁽²⁾,𝜃⁽²⁾,𝑝⁽²⁾,𝐵⁽²⁾, respectively. If 𝜆⁽¹⁾≤𝜆⁽²⁾,𝜃⁽¹⁾≥𝜃⁽²⁾,𝑝⁽¹⁾≤𝑝⁽²⁾,𝐵⁽¹⁾(𝑥)≤_𝐿𝐵⁽²⁾(𝑥), then 𝜑_𝑛⁽¹⁾≤_𝐿𝜑_𝑛⁽²⁾.

Proof. By Lemma 3.4, we have 𝛽⁽¹⁾(𝜆⁽¹⁾(1−𝑧))≥𝛽⁽²⁾(𝜆⁽²⁾(1−𝑧)).
Moreover, one has 𝑝⁽¹⁾≤𝑝⁽²⁾⇒𝑝⁽¹⁾+𝑝⁽¹⁾𝑧≥𝑝⁽²⁾+𝑝⁽²⁾𝑧,forall𝑧∈[0,1].
This implies that 1−𝑝⁽¹⁾+𝑝⁽¹⁾𝑢𝛽⁽¹⁾𝜆⁽¹⁾−𝜆⁽¹⁾𝑢𝑝⁽¹⁾+𝑝⁽¹⁾𝑢𝛽⁽¹⁾𝜆⁽¹⁾−𝜆⁽¹⁾𝑢≤−𝑢1−𝑝⁽²⁾+𝑝⁽²⁾𝑢𝛽⁽²⁾𝜆⁽²⁾−𝜆⁽²⁾𝑢𝑝⁽²⁾+𝑝⁽²⁾𝑢𝛽⁽²⁾𝜆⁽²⁾−𝜆⁽²⁾𝑢−𝑢. (6.2) Besides, 𝜆⁽¹⁾≤𝜆⁽²⁾ and 𝜃⁽¹⁾≥𝜃⁽²⁾⇒(𝜆⁽¹⁾/𝜃⁽¹⁾)≤(𝜆⁽²⁾/𝜃⁽²⁾) and thus 𝜆⁽¹⁾𝜃⁽¹⁾¹_𝑧1−𝑝⁽¹⁾+𝑝⁽¹⁾𝑢𝛽⁽¹⁾𝜆⁽¹⁾−𝜆⁽¹⁾𝑢𝑝⁽¹⁾+𝑝⁽¹⁾𝑢𝛽⁽¹⁾𝜆⁽¹⁾−𝜆⁽¹⁾𝑢𝜆−𝑢𝑑𝑢≤⁽²⁾𝜃⁽²⁾¹_𝑧1−𝑝⁽²⁾+𝑝⁽²⁾𝑢𝛽⁽²⁾𝜆⁽²⁾−𝜆⁽²⁾𝑢𝑝⁽²⁾+𝑝⁽²⁾𝑢𝛽⁽²⁾𝜆⁽²⁾−𝜆⁽²⁾𝑢−𝑢𝑑𝑢. (6.3) Therefore −𝜆exp⁽¹⁾𝜃⁽¹⁾¹_𝑧1−𝑝⁽¹⁾+𝑝⁽¹⁾𝑢𝛽⁽¹⁾𝜆⁽¹⁾−𝜆⁽¹⁾𝑢𝑝⁽¹⁾+𝑝⁽¹⁾𝑢𝛽⁽¹⁾𝜆⁽¹⁾−𝜆⁽¹⁾𝑢−𝜆−𝑢𝑑𝑢≥exp⁽²⁾𝜃⁽²⁾¹_𝑧1−𝑝⁽²⁾+𝑝⁽²⁾𝑢𝛽⁽²⁾𝜆⁽²⁾−𝜆⁽²⁾𝑢𝑝⁽²⁾+𝑝⁽²⁾𝑢𝛽⁽²⁾𝜆⁽²⁾−𝜆⁽²⁾𝑢.−𝑢𝑑𝑢 (6.4) By combining this latter inequality with the inequality: 𝑝⁽¹⁾+𝑝⁽¹⁾𝑧≥𝑝⁽²⁾+p⁽²⁾𝑧, we get 𝜙⁽¹⁾(𝑧)≥𝜙⁽²⁾(𝑧)forall𝑧∈[0,1], which means the stochastic inequality {𝜑_𝑛⁽¹⁾}≤_𝐿{𝜑_𝑛⁽²⁾}.

Theorem 6.2. For any 𝑀/𝐺/1 retrial queue with classical retrial policy and Bernoulli feedback the distribution 𝜑_𝑛 is less relative to the Laplace ordering than the distribution with the generating function −𝜆𝑝+𝑝𝑧exp𝜃¹_𝑧1−𝑝+𝑝𝑢𝑒^{𝜆𝛽1(𝑢−1)}𝑝+𝑝𝑢𝑒^{𝜆𝛽1(𝑢−1)},−𝑢𝑑𝑢 (6.5) and if 𝐵(𝑥) is of class ℒ then the distribution 𝜑_𝑛 is greater relative to the ordering ≤_𝐿 than the corresponding distribution in the 𝑀/𝑀/1 queue with classical retrial policy and Bernoulli feedback.

Proof. Consider an auxiliary 𝑀/𝐷/1 and 𝑀/𝑀/1 retrial queues with classical retrial policy and Bernoulli feedback with the same arrival rates 𝜆, probability 𝑝, retrial rates 𝜃, and mean service times 𝛽₁.
Since 𝐵(𝑥) is always less, relative to the ordering ≤_𝐿, than a deterministic distribution with the same mean value, based on Theorem 6.1 we obtain the stated result.
If 𝐵(𝑥) is of class ℒ then 𝐵(𝑥) is greater relative to the ordering ≤_𝐿 than the exponential distribution with the same mean, based on Theorem 6.1 we can guarantee the second inequality.

7. Conclusion and Further Research

In this paper, we prove the monotonicity of the transition operator of the embedded Markov chain relative to strong stochastic ordering and increasing convex ordering. We obtain comparability conditions for the distribution of the number of customers in the system. Inequalities are derived for conditional distribution of the stationary queue given that the server is idle. The obtained results allow us to place in a prominent position the insensitive bounds for both the stationary distribution and the conditional distribution of the stationary queue of the considered model.

Monotonicity results are of importance in robustness analysis: if there is insecurity on the input of the model, then our order results provide information on what kind of deviation from the nominal model to expect. Moreover, in gradient estimation one has to control the growth of the cycle length as function of a change of the model. More precisely, the results established in this paper allow to bound the measure-valued derivative of the stationary distribution where the derivative can be translated into unbiased (higher order) derivative estimators with respect to some parameter (e.g., arrival rate (𝜆) or retrial rate (𝜃) parameter). Such bounds can be used to derive information on the speed of convergence of the gradient estimator. Finally, under some conditions (order holds in the strong sense), those results imply a fast convergence of the gradient estimator of the stationary distribution [14–16].