This paper considers a retrial queueing model where a group of guard channels is reserved for priority and retrial customers. Priority and normal customers arrive at the system according to two distinct Poisson processes. Priority customers are accepted if there is an idle channel upon arrival while normal customers are accepted if and only if the number of idle channels is larger than the number of guard channels. Blocked customers (priority or normal) join a virtual orbit and repeat their attempts in a later time. Customers from the orbit (retrial customers) are accepted if there is an idle channel available upon arrival. We formulate the queueing system using a level dependent quasi-birth-and-death (QBD) process. We obtain a Taylor series expansion for the nonzero elements of the rate matrices of the level dependent QBD process. Using the expansion results, we obtain an asymptotic upper bound for the joint stationary distribution of the number of busy channels and that of customers in the orbit. Furthermore, we develop an efficient numerical algorithm to calculate the joint stationary distribution.
In this paper, we consider multiserver retrial queues with guard channels for priority and retrial customers. Retrial queues are characterized by the fact that a blocked customer repeats its request after a random time. During retrial intervals, the customer is said to be in the orbit. This type of queueing models is widely used in modelling and performance analysis of communication and service systems, especially in cellular networks [
The guard channel concept has been extensively used in communication systems [
The analysis of multiserver retrial queues with infinite orbit size is challenging due to the fact that the underlying Markov chain is state dependent because the retrial rate is proportional to the number of customers in the orbit. Thus, even for the fundamental model with one type of traffic and without guard channels, the stationary distribution is expressed in terms of simple functions for only some special cases, that is, one or two servers [
This motivates us to consider a new model with both retrials and guard channels for which we explore new analytical and numerical results. From the modelling point of view, the novelty is the priority given to retrial customers. It should be noted that retrial customers are treated the same as normal customers in [
The stationary distribution of a level dependent QBD process can be expressed in terms of a sequence of rate matrices [
The main contribution of our paper is threefold. First, using a censoring technique and a perturbation method, we obtain the Taylor series expansion for the rate matrices in terms of the number of customers in the orbit. Our formula is general in the sense that we can obtain the expansion with arbitrary number of terms. This was not reported in Liu and Zhao [
The rest of our paper is organized as follows. Section
In this paper, we consider a queueing model with two types of customers (types 1 and 2). There are
In this paper, we restrict ourselves to the case of one guard channel; that is,
From a theoretical point of view, the assumption that retrial customers (both normal and priority) have the same priority significantly simplifies the analysis. This is because if retrial customers keep their initial priority, we need to distinguish two types of retrial customers for which we should have two orbits for priority and normal customers.
Let
The proof is presented in Appendix
It is easy to see that
Let
Lemmas
It is easy to see that the first
It should be noted that, in comparison with a previous version [
In Section
One has
This lemma follows from the fact that the following matrix represents the infinitesimal generator of the ergodic Markov chain
For case
Furthermore, substituting these explicit expressions into (
It should be noted that the explicit results in Corollary
In this section, we derive the Taylor series expansion for all the nonzero elements of the rate matrices. In particular, we find the Taylor series expansion of
In this section, Lemma
In this section, we find the Taylor expansion for nonzero components. The basic idea in a perturbation approach is that the coefficient of
One has the one-term series expansion (
The technical details are provided in Appendix
The series expansion formulae (
The technical details are provided in Appendix
The nonzero elements of
The technical details are provided in Appendix
In this section, we present an asymptotic upper bound for the stationary distribution. To this end, we use Lemmas
For a square matrix
For integers
In [
One defines
The proof uses Lemmas
It follows from
For a sufficiently large
Thus, for the parameters that satisfy Lemma
One has
From
In this section, we propose a computational algorithm for the stationary distribution of our model extending that proposed by Phung-Duc et al. [
Due to Lemma
It should be noted that the computation of
We define the function
It is easy to see that
For arbitrary
The technical details are provided in Appendix
For arbitrary
This lemma can be proved using the same technique as in Lemma
Solution
Computation of
Sequence
We prove Theorem
Using Theorem
Tran-Gia and Mandjes [
In this section, we present an algorithm for computing the rate matrices and then a procedure for the computation of the stationary distribution. Algorithm
Compute
Compute
Compute
In Algorithm
However, since
Let
Using this result, we set the truncation point as follows:
We derive blocking probabilities as performance measures. In our model, priority (type 1) and retrial customers are blocked when all the servers are occupied while normal (type 2) customers are blocked when at least
In this section, we show some numerical examples. In particular, in Section
The rate matrix is calculated using Algorithm
First, we present some numerical examples to show the effectiveness of the Taylor series expansion. Tables
Relative error for
( |
One term | Two terms | Three terms |
---|---|---|---|
0.1 | 0.0051053401 | 0.0003425140 | 0.0000228094 |
0.2 | 0.0086100661 | 0.0006446694 | 0.0000491957 |
0.3 | 0.0120849796 | 0.0009702635 | 0.0000821267 |
0.4 | 0.0155304303 | 0.0013188638 | 0.0001219509 |
0.5 | 0.0189467632 | 0.0016900430 | 0.0001690102 |
0.6 | 0.0223343192 | 0.0020833798 | 0.0002236397 |
0.7 | 0.0256934342 | 0.0024984580 | 0.0002861679 |
0.8 | 0.0290244403 | 0.0029348670 | 0.0003569166 |
0.9 | 0.0323276648 | 0.0033922015 | 0.0004362009 |
Relative error for
( |
One term | Two terms | Three terms |
---|---|---|---|
0.1 | 0.0004109454 | 0.0000030629 | 0.0000000407 |
0.2 | 0.0008055339 | 0.0000063695 | 0.0000000804 |
0.3 | 0.0011997344 | 0.0000099832 | 0.0000001323 |
0.4 | 0.0015935474 | 0.0000139034 | 0.0000001977 |
0.5 | 0.0019869735 | 0.0000181294 | 0.0000002747 |
0.6 | 0.0023800133 | 0.0000226607 | 0.0000003638 |
0.7 | 0.0027726674 | 0.0000274965 | 0.0000004657 |
0.8 | 0.0031649363 | 0.0000326363 | 0.0000005807 |
0.9 | 0.0035568205 | 0.0000380794 | 0.0000007095 |
Relative error for
( |
One term | Two terms | Three terms |
---|---|---|---|
0.1 | 0.0004109342 | 0.0000030754 | 0.0000000215 |
0.2 | 0.0008055116 | 0.0000063974 | 0.0000000500 |
0.3 | 0.0011997010 | 0.0000100293 | 0.0000000863 |
0.4 | 0.0015935030 | 0.0000139704 | 0.0000001309 |
0.5 | 0.0019869182 | 0.0000182201 | 0.0000001843 |
0.6 | 0.0023799470 | 0.0000227778 | 0.0000002472 |
0.7 | 0.0027725901 | 0.0000276429 | 0.0000003200 |
0.8 | 0.0031648480 | 0.0000328146 | 0.0000004033 |
0.9 | 0.0035567214 | 0.0000382924 | 0.0000004976 |
Relative error for
( |
One term | Two terms | Three terms |
---|---|---|---|
0.1 | 0.0000401092 | 0.0000000304 | 0.0000000000 |
0.2 | 0.0000800545 | 0.0000000640 | 0.0000000001 |
0.3 | 0.0001199958 | 0.0000001007 | 0.0000000001 |
0.4 | 0.0001599331 | 0.0000001406 | 0.0000000002 |
0.5 | 0.0001998664 | 0.0000001837 | 0.0000000003 |
0.6 | 0.0002397957 | 0.0000002300 | 0.0000000004 |
0.7 | 0.0002797210 | 0.0000002795 | 0.0000000005 |
0.8 | 0.0003196424 | 0.0000003321 | 0.0000000006 |
0.9 | 0.0003595598 | 0.0000003880 | 0.0000000007 |
We observe that the Taylor series expansion gives a good approximation in the sense that the relative error is quite small. The relative errors for case
Furthermore, we observe that relative error in Tables
Figure
Figure
We use the following fixed parameters
Blocking probability versus the number of servers.
An important observation is that all the curves are asymptotically linear when the number of servers is large. Asymptotic analysis for the case of large number of servers may be the topic of any future research. In this direction, Avram et al. [
In this paper, we have introduced a new queueing model with a guard channel for retrial and priority customers. The new queueing model is formulated using QBD process which possesses a sparse structure allowing an efficient numerical algorithm and the Taylor series expansion for all the nonzero elements of the rate matrices. We have also derived an asymptotic upper bound for the joint stationary distribution. Numerical results have revealed that the upper bound can be further improved. Future work includes finding the exact asymptotic formulae for the joint stationary distribution.
We prove the sufficient condition in Lemma
Let
Next, we prove that, except for
We prove that, for
Arranging (
We prove Lemma
We prove Theorem
Next, assuming that Theorem
Let
For arbitrary
Rewriting this equation we obtain
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank Professor Peter G. Taylor for some useful comments which help to improve the presentation of the paper. Tuan Phung-Duc was supported in part by JSPS KAKENHI Grant no. 26730011.