Mathematical modeling on minimizing the waiting time in (i,j) preemptive priority queueing system

In this paper, we analyze the waiting time in two stage preemptive priority queueing system. The service time considered here is k-phase Erlang distribution. We have assign the priority server to the emergency customer without affecting the regular priority unit. It is possible only if the servers are independent. In this case, we derive the steady state probability of the waiting time customers in ordinary and priority unit. A numerical example is included.


1.Introduction
Due to some special situations the queueing model having some different type of customers. In general, priority system having two types as follows: (i) PP-Pre-emptive Priority (ii) NPP-Non -Preemptive Priority. The type (i) is a higher priority customer not interrupted to the entire service time of lower priority customer and otherwise it is called type (ii). Due to the situation the discipline of the priority model is not proper. However, in a situation we separate the service requirements are parallel for each class to improve the system performance level and minimize the waiting time. Kleinrock was first proposed the algorithm and analyzed the discipline of priority customer. Latter many researchers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] have, followed by Kleinrock and proposed different algorithms for priority queueing system and also obtained the system performance. Here, we derive the waiting time probability of priority unit customer and derive the queue length of ordinary unit. In this connection, we consider the system in a general queueing system with k-phase, multi server; arriving customer considered in Poisson model and service time follows two units: (i) OU-Ordinary Unit (postponed) (ii) Priority unit (misplaced).

Model description on two server priority Queue
Consider the queueing system in which the service time is not available for immediate because the number of servers is busy and new arrivals are not allowed to begin the service even some channels are idle. Suppose the arrival expects to get an immediate service to a unit, it may either wait for service or else leave the system. Once a unit enters the service, it is not pre-empted. If ij NN  the units of the classes 'i & j' are served according to the discipline, 'i' having non-pre-emptive priority over 'j' if i < j. Finally, a class of units queues according to the FIFO discipline. Thus, for a twoclass priority process, the discipline of the break-off priority channels is specified as follows: N channels are occupied then the services are taken whether priority unit or ordinary unit. Otherwise, numbers of channels are occupied in greater than 1 N the services given only be the priority. But it is not possible to pre-empt the service of low priority units in order to a higher priority unit.

Probability of waiting time for the Priority Unit
Here, we assumed the priority unit is lost when all the channels are busy. On the other hand the ordinary units wait in the queue when they arrive and find 1 N or more channels are busy. To identify the state of the system will require the number of busy channels and the number of ordinary units waiting in the queue. Let ) , ( 2 n m q denotes the steady state probability that there are 2 m ordinary units present in the queue and 'n' channels are busy. The representativeness of the transition rates into out of the states of the system. The difference equation connecting the probabilities defined in (1) and (2) it can be written as from which we obtain the diagram.
is the probability of all the channels being idle.
And hence Thus, (*, 0) q lies between 1 0 n N  and (*, ) qn lies between 1 N n N are completely determined. The equation (6) represents the loss of priority units and steady state probability that the ordinary unit waits is given by 1

3.Derivation of Queue Length for the Ordinary Unit
If the number of 1 NN  the priority unit is lost by the proportion of time N busy channel q(N).

Consider the arrival unit in both 12
    and the customer begin the service more than one (n>1). The steady state probability that the n servers are busy, the probability of all the servers idle denoted by ( 2) e . From the diagram the approximate equation of steady state is as follows: The steady state probability 1 l that a probability unit is lost is given by the proportion of time N channels are busy, and is therefore given by () qN for an ordinary unit becomes

4.Numerical Example
Consider a Breast Cancer center with a branch in a famous city. There is only one Chief Doctor who treats the patients after their consultation with junior doctors in preliminary 3-phases. The arrival pattern of the patients is assumed to have an Erlang 3-distribution with arrival rate λ. The service time of the patients is assumed to have an exponential distribution with service rate μ. Consider the range of the frequencies of transition from state i to state j Find the probability of ordinary unit waiting time and also find the queue length of priority unit.

5.Conclusion
We consider the two stage k-phase priority queueing system for the purpose of minimizing the queue length. In this regard, derived the Waiting time probability of priority unit and Queue Length of Ordinary Unit. Table 1and 2 has represented to minimize waiting time of length and the probability of waiting time in ordinary unit are very close to zero. So, our proposed method is easy to reduce the queue waiting time and also it's helpful to the decision making problem.