Retrial queues have been extensively studied, since they arise in various systems such as telephone switching systems, telecommunication, call centers and computer networks. The characteristic feature of retrial queue is that when arriving customers find the server is unavailable, then the customer makes a new attempt to get service after a random
For a complete survey on retrial queues, refer the work of Artalejo
There are two common situations in priority discipline, one is preemptive and other is non-preemptive. In the case of preemptive, the customer with higher priority is allowed to receive service immediately even if a customer with lower priority is already in service; that is, the service of lower priority customer should be preempted or stopped and to be restart again after the service of high priority customer. In the second case of priority discipline which is called non-preemptive, the higher priority customer goes to the front of the queue but cannot get service until the customer presently in service has completed his service, even though the customer in service has a lower priority.
An obvious example of such a situation deals with the classification of patients arriving at an emergency room of a hospital. Another example appears in a health care system in which the patients who require an immediate surgery or transplant have provided a priority over the patients, who has been otherwise waiting in a line. In many communication systems, a priority is given to certain classes or calls to improve the grade of service over other classes. In a wireless communication system, which delivers a wide variety of services, priority rule is often followed. For example: voice calls being delay sensitive are required to have a higher priority than data calls. While it is true that priority systems reduce the waiting times of the higher priority customers, they also necessarily increase the waiting times of the lower priority ones.
Nair et al.
Pegden & Rosenshine
Many other researchers already worked on the concept of retrial queues with non preemptive priority system but the major difference is that they deal with total number of customers in the system instead of considering the two-dimensional state retrial queueing system in the form of (i,j). Here ‘i’ represent total number of arrivals in the system and ‘j’ represent total number of departures from the system by time t. One similar study “two-state retrial queueing system with priority customers” has also been presented. In this study, authors found the time dependent probabilities of exact number of arrivals and departures from the orbit instead of finding the probabilities from the system. In my present study, the time-dependent probabilities of exact number of arrivals (primary arrivals, arrivals in high priority queue, arrivals in low priority queue) in the system and exact number of departures (primary departures, departures from high priority queue, departures from low priority queue) from the system by a given time have to be found. The present work is better than the previous research because here we considered the units from the system (including orbit) instead of considering the units only from the orbit and also considered the whole system including orbit gives finer results comparatively to the results consider from the orbit.
This investigation considers a general comparison of other systems in the sense that we have combined different characteristics together covering (i) non-preemptive priority discipline (ii) retrial queueing system (iii) two-dimensional state. The model proposed by us by examining the three above factors simultaneously provided a superior model contrasting the other queueing models.
Singla and Kaur
The summary of this paper is as follows: Section 2, deals with model description, notations used, mathematical formulation and difference differential equations of the model. Section 3, elucidates the transient state solution of our model. Section 4, demonstrates and verify some important results along with some special cases. Numerical solution and graphical representation are given in section 5 with some busy period probabilities of system and server. In section 6, concluding remarks are given.
A two-state M/M/1 retrial queueing model with priority subscribers is considered. A primary customer, who enters into the queueing system arrives according to a Poisson process with rate
Laplace transformation
The Laplace inverse of
where,
P(p)=
Q(p) is a polynomial of degree <
If
The Laplace inverse of
If
An M/M/1 queue is a stochastic process whose state space is the set
For distribution of arrivals, service times and retrials we use the following assumptions and notations:
The primary arrivals follow a Poisson distribution with parameter
The high priority arrival calls and low priority arrival calls also follow a Poisson distribution with parameter
The high priority calls and low priority calls repeated their attempt to get service in a Poisson distribution with parameter
Service times are exponentially distributed with parameter
The stochastic process involved viz. arrivals of units, departures of units and retrials are statistically independent.
also
Initially
Using the Laplace transformation
in the equations (2.1) - (2.5) along with the initial conditions, we have
Solving equations (2.6) to (2.10) recursively, we have
Taking the Inverse Laplace transform of equations (3.1) to (3.14), we have
From the abstract solution of two-dimensional state model, it is verified that the sum of all possible probabilities is one. On taking summation over
on taking inverse Laplace transformation it gives
Which is a verification of obtained results.
To convert two-dimensional state model into single state model, probability
The probability of exactly
When the server is free, it is defined by probability
Where
When the server is busy, it is defined by probability
Where
Using the above definitions, from the equations (2.1) to (2.5), the set of equations in statistical equilibrium are:
(b) Again assuming
these equations coincide with result (1.5) and (1.6) of Falin & Templeton [4].
In this section, some numerical results that describe the model under study are examined. Numerical results are generated using MATLAB programming for the cases
Various probabilities are plotted against time
Figure 2 shows relative changes in probabilities
Probabilities
To study the effect of traffic intensity (customers are arriving per unit service time) on different probabilities of the model, the data of various probabilities are generated for different values of
In
The busy period analysis plays a vital role in understanding various operations taking place in any queueing system. A busy period is initiated with the arrival of a customer who finds the system empty and ends when the next system becomes free. In this section, we discuss some numerical results about the busy period distribution of the server and busy period distribution of the system.
The probability when the server is busy is given by
P (Server is busy) =
The probability when the system is busy is given by
P (System is busy) =
The numerical results are generated using MATLAB programming. The probability when the system is busy and the probability when the server is busy are presented in
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0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0.1720 |
0.3070 |
0.4129 |
0.2003 |
0.3624 |
0.4930 |
2 |
0.2319 |
0.3889 |
0.4903 |
0.3061 |
0.5278 |
0.6827 |
3 |
0.2614 |
0.4164 |
0.4924 |
0.3767 |
0.6262 |
0.7753 |
4 |
0.2786 |
0.4178 |
0.4608 |
0.4274 |
0.6820 |
0.8054 |
5 |
0.2877 |
0.4008 |
0.4129 |
0.4628 |
0.7013 |
0.7866 |
6 |
0.2898 |
0.3709 |
0.3591 |
0.4843 |
0.6886 |
0.7324 |
7 |
0.2856 |
0.3331 |
0.3057 |
0.4929 |
0.6501 |
0.6563 |
In
The probability when the system is busy is plotted in
The factor “two-state” makes our model more realistic and quantified. Discussion in this article enriches modern queueing theory, and has wide use in practical questions. As in computer communication net, in order to offer multi-layer quality of service for different kind of customer, priority control is necessary. The presence of retrial customers can radically change the behavior of a queueing system. This model developed two features simultaneously including retrial and priority, which makes our results applicable to more versatile congestion situations encountered in computer and communication systems, production and manufacturing systems and distribution and service sectors.