Stochastics and Statistics
Analysis of the BMAP/G/1 retrial system with search of customers from the orbit

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Abstract

We consider a single server retrial queuing model in which customers arrive according to a batch Markovian arrival process. Any arriving batch finding the server busy enters into an orbit. Otherwise one customer from the arriving batch enters into service immediately while the rest join the orbit. The customers from the orbit try to reach the service later and the inter-retrial times are exponentially distributed with intensity depending (generally speaking) on the number of customers on the orbit. Additionally, the search mechanism can be switched-on at the service completion epoch with a known probability (probably depending on the number of customers on the orbit). The duration of the search is random and also probably depending on the number of customers in the orbit. The customer, which is found as the result of the search, enters the service immediately if the server is still idle. Assuming that the service times of the primary and repeated customers are generally distributed (with possibly different distributions), we perform the steady state analysis of the queueing model.

Introduction

A retrial queueing system is characterised by the requirement that customers encountering all servers busy will have to leave the area and join a group of unsatisfied customers, called orbit. These arise frequently in the stochastic modelling of telecommunications, computer systems, stock and flow and so on. Review of retrial queueing literature could be found in [1], [2], [14], [15], [23]. A number of application of retrial queues in science and engineering can be found in [19]. In the retrial setup, each service is preceded and followed by the server(s) idle time because of the ignorance about the status of the server(s) and orbital customers by each other.

We are interested in designing retrial queues that reduces the server(s) idle time and achieve this by the introduction of search of orbital customers immediately after a service completion (with search we associate a probability). Search for orbital customers was introduced in [3]. It may be recalled that in [22] the authors examined classical queue with search for customers immediately on termination of a service. An example for search of orbital customers is the following: consider an inventory where customers requiring the item arrive for purchase. Incase the item is out of stock at a demand epoch, such customers, after recording their name in a register, proceed to the orbit (can be their own residence). On replenishment the management scans through this register and select persons (according to some priority), to provide the item.

Yet another example is the repair-service with search of customers: the job-shop keeps a register of customers who are forced to leave the system since they encountered a busy server at the time of arrival. On completing a service the server decides to have the next service by picking up an unsatisfied (orbital) customer with probability pj, where j is the number of customers on the orbit. Simultaneously, the customers from the orbit try to reach the server themselves and there is a competition between primary and orbital customers and search mechanism. If the search mechanism is not switched (with probability qj=1−pj), there is a competition only between primary and secondary customers for getting into the counter for the next service.

This paper generalizes the result in [3] introducing a search time, two types of services to customers (primary/orbital) and by assuming the arrival process to be batch Markovian process. The batch Markovian arrival process (BMAP) suits well for modelling the correlated bursty traffic in the modern communication networks. Approximation of such flows in terms of the stationary Poisson process can cause huge errors in evaluation of performance characteristics. Chakravarthy in [5] provides a review of queueing models with batch Markovian arrivals. Retrial models with BMAP arrivals have been investigated in [4], [6], [7], [9], [10], [11], [18].

In Section 2, the mathematical model under consideration is described. In Section 3, the stationary distribution of the embedded Markov chain at the service completion epochs is calculated. Section 4 contains formulas for calculating the arbitrary time orbit length stationary distribution and some performance characteristics of the model. In Section 5, the modification of results (previously obtained for negligible search time) to the case of random search time is given. Section 6 gives the numerical illustration of the model presented and way for calculating the stationary distributions. The last section contains some concluding remarks and probable ways of extension of results to the cases of semi-Markovian service process and state dependent service time distributions.

Section snippets

The mathematical model

We consider a single server queueing system in which the arrivals occur according to a BMAP. The BMAP, a special class of tractable Markov renewal process, is a rich class of point processes that includes many well-known processes such as Poisson, PH-renewal processes, and Markov-modulated Poisson process. One of the most significant features of the BMAP is the underlying Markovian structure and fits ideally in the context of matrix-analytic solutions to stochastic models. The matrix-analytic

The stationary distribution of the embedded Markov chain

For the sake of simplicity, assume that the search mechanism is always switched-on at service completion epochs (i.e., pi=1, i>0). Later we will relax this assumption.

Let tn denote the nth service completion epoch, in be the number of customers in the orbit and νn be the state of the BMAP process νt at tn+0. It is clear that the process ξn={in,νn}, n⩾1, is a two-dimensional Markov chain with state space in⩾0, νn=1,…,m.

In the sequel we need the following auxiliary matrices. DefineFi=∫0e(D0−αiI)t

The stationary distribution of the queue at arbitrary time

Denote by p(i,r), i⩾0, 0⩽r⩽2, the steady state probability vector that at an arbitrary time there are i customers in the system, and the current service is in rth mode. Note that r=0 corresponds to the case when the server is idle. The following theorem gives an expression for the steady state probabilities.

Theorem 2

The stationary probability vectors p(i,r) are calculated as follows:p(i,0)=τ−1πiFi,i⩾0,p(i,1)=τ−1l=1iπlΦlΩi−l(1),i>0,p(i,2)=τ−1l=0i−1πlFlk=1i−lDkΩi−l−k(2),i>0,where the matrices Ωm

The case of randomized beginning of a search

To reduce the size of formulae above, we temporarily assumed that the search begins at any service completion epoch (when the orbit is not empty). Recall now that we assumed earlier that the search begins with probability pi if i customers stay in the orbit. In the opposite case the system behaves as the usual retrial system and the service begins after the arrival of a primary batch or a retrial. Analyzing the behavior of embedded Markov chain in this case we easily see that the above

Numerical examples

In this section, we present some numerical examples. The numerical realization of proposed algorithms for calculating the embedded and arbitrary time stationary distributions is implemented relatively easily only if we have already the existing infrastructure suitable for calculating, e.g., the matrices like Ωl(r), Ωl(r), l⩾0, r=1,2, Y(1), Y(1), G, etc. Such a structure is provided by the software “SIRIUS+” (see, e.g., [13]).

We consider the BMAP with representation {Dk} given by the matrices:D0

Concluding remarks

In this paper we considered a single server retrial queuing model in which customers arrive according to a BMAP. The services are offered in two modes. In mode 1, the customers are served after staying in the orbit while in mode 2 the customer is served in case it has a luck to meet a free server upon arrival. In addition to the usual way of reaching the server due to a customer retrial, we assume that, at the service completion epoch, the server can start a search of a customer in the orbit

Acknowledgements

Research of A. Krishnamoorthy is supported by UGC (India), Research award F.30-97/98/SA-III.

Research of V.C. Joshua is supported by UGC (India) under Faculty Improvement Programme F.FIP99/SWRO/UGC/KLMG002.

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