TIME DEPENDENT RETRIAL QUEUEING MODEL WITH ORBITAL SEARCH UNDER NON-PREEMPTIVE PRIORITY SERVICES

Transient solution of Single server Non-preemptive priority Retrial model with orbital search is studied using eigenvalues and eigenvectors. In this model, customers are arriving in Poisson process. Arrival rates of low priority and high priority customers are respectively λ1 and λ2. The service times for low and high priority customers follow exponential distribution with parameters μ1 and μ2 respectively. Customer finding the system busy, on arrival, goes to the orbit and form a virtual queue. In this model of orbital search, when the server becomes free, he has two options. Either the server search for the customer in the orbit with probability p to provide service for a customer in the orbit or remains idle with the probability 1-p. Whenever the server is free, customer from orbit try for service, under classical retrial policy with rate σ which follows Poisson process. In this paper, transient solution of average number of customers in the orbit and high priority queue, the probability of server being idle, the probabilities of server being busy with low and high priority customers for various values of λ1, λ2, μ1 , μ2, σ, p and t are estimated.


INTRODUCTION
Retrial is a branch inherited from Queueing Theory. Customers finding the server busy, on arrival to the system, form a virtual queue with the intention to try for service after some random time.
This virtual queue is termed as Orbit. Customers in the orbit retry independently of each other for service by following one of the retrial policy such as Classical retrial policy, Constant retrial policy and Linear retrial policy. Though Retrial Queueing theory was initiated in 1953, the momentum was gained only in the eighties. Falin, Templeton, Artalejo, Choi, Le Gall, Neuts, Stepanov and Yang were the major contributors to the Theory of Retrial Queue. Review of Literature [1], [2], [10], [12], [28] can be found from Surveys and Bibliographies. Falin etal. studied the Single server Retrial queue with priority services.
One of the main objective of Queueing theory is to minimize the idle time of the server. In Retrial Queueing Theory, when the server becomes free, he remains idle until either a low or a high priority customer arrives to the system or a customer from the orbit retry for the service. To minimize this idle time of the server, a new approach is advocated. The technique is that whenever the server becomes free, either he voluntarily search for the customer in the orbit to provide the service with probability p or remain idle with probability 1-p. This process is called Orbital Search. Artalejo [3] introduced Orbital search and initiated the study in 2002. Dudin, Krishnamoorthy, Joshua and Deepak [8] [9] [16]  In many of the applications, Transient solution to a Queueing Problem is considered more relevant than the steady state. Transient solutions to Queueing problem were initially explored by Clarke [5], Saaty [19] and Tacaks [27] in the middle of the Twentieth century. Nearly after three decades, the interest on study of transient solution was evinced by Parthasarathy [17], and Krishnakumar [15].
Search of transient solution to Retrial Queueing problem is more difficult and much complicated than that of Queuing Problem. Due to the presence of differential equations, Analytical solution to Transient Retrial Queueing problem is hard to manage. The alternative is to look for Numerical solution. The transitive solution to M/M/1 Retrial queueing system is evolved using Eigen values and Eigenvectors. In this paper, Numerical solution to Transient M/M/1 Retrial Queueing model with non-preemptive priority service and orbital search is obtained using eigenvalues and eigenvectors.

MODEL DESCRIPTION
The model taken up for study is Single server Retrial Queueing system with orbital search under non-preemptive priority service. In this priority model, the arrival of low and high priority customers follow Poisson distribution with arrival rates respectively λ1 and λ2. If the server is idle and a low priority customer arrives, then the server provides service to the low priority customer and the customer leaves the system after he gets the service. On the other hand, when a low priority customer, finds the server busy on his arrival, goes to the orbit with the objective to try for service independently after some time. The rate of retrial of customers from orbit follows Poisson Process with retrial rate σ.
The time taken by the server to provide service for low and high priority customers follow exponential distribution. The rates of service to low and high priority customers are respectively μ1 and μ2. An arriving high priority customer to the server may find himself in one of the following situations When high priority queue is empty 1. When the server is idle, he will get the service immediately and leaves the system.
2. When the server is busy with a low priority customer, as the model is Non-preemptive priority model, the high priority customer wait for the server to complete the service for the low priority customer and then get the service done and leaves the system. 3279 TIME DEPENDENT RETRIAL QUEUEING MODEL 3. When the server is busy with high priority customer, wait for the service to be completed and gets the service and leave the system. When high priority queue is nonempty 4. Joins the high priority queue.
In this model of Orbital search, in order to minimize the idle time of the server, whenever the server becomes free, he has two options. Either he provides service for a customer in the orbit with chance p or remain idle with chance 1-p. The time duration to call the low priority customer for service is considered as negligible.

RETRIAL POLICY
Customers from the Orbit try to get the service under Classical Retrial Policy. In this policy, Customers from the orbit try independently to get service from the server. The random time to get selected for the service is exponentially distributed with rate σ. In the interval (t, t +∆t), the probability of retrial, when there are n customers in the orbit, is nσ∆t + O(∆t).

RANDOM PROCESS
Let N(t), P(t) and S(t) be the random variables representing, the number of customers in the orbit at time t, the number of high priority customers in the queue at time t and the state of the server at time t respectively. The random process is described as <N(t) , P(t), S(t) >.

TRUNCATION
As the set of State spaces is infinite, the infinitesimal generator matrix will not be a matrix of finite order. Matrix of finite order is desired to facilitate the computation and it is obtained by restricting the number of customers in the high priority queue to s and the number of customers in the orbit needs to be truncated to M so that the probability that will be lost is negligible. In this model, open truncation is applied to overcome the difficulty.

TRANSITIONS
The transitions can be broadly categorized into three cases depending on the state from which it is being transitioned. They are

GOVERNING DIFFERENTIAL EQUATIONS
Let Pijk(t) stands for the Probability at time t for N(t) = i, P(t) = j and S(t) = k. Then from the transition tables Table 1,2,3, the Chapman -Kolmogorov Difference Differential equations for the Single server retrial queueing model with non-preemptive priority service with orbital search are obtained as shown in system of equations (1). The equations can be written in matrix form as

TRUNCATED GENERATOR MATRIX
The truncated generator matrix is The matrices ij A can be obtained from the system of Chapman -Kolmogorov equations.
Here 0 The solution of the system of equations ( )

DESCRIPTION OF COMPUTATIONAL METHOD
To obtain the solution, the following sequence is used 1. Orbit size is truncated to a sufficiently large value of the number of customers in the orbit, say M, so that the loss probability is small.

2.
Generator matrix Q is a square matrix of order (2s+3)(M+1). The time dependent probabilities can be written as

The probability that the orbiting customer is blocked at time t
Blocking Probability = where MNCO is the mean number of low priority customers in the Orbit and MPQL is the mean number of high priority customers in the steady state.

CONCLUSION
In this paper, with the intention of reducing the idle time of the server, Orbital search in the case of Single Server Retrial queueing system with non-preemptive priority is studied. Transient solution to the M/M/1 Retrial Queueing system is obtained using eigenvalues and eigenvectors.
System performance measures and Probability distributions of customers and System performance measures are determined.