CHAPTER-5 PERFORMANCE ANALYSIS OF AN M/M/1/K QUEUE WITH PREEMPTIVE PRIORITY

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1 CHAPTER-5 PERFORMANCE ANALYSIS OF AN M/M//K QUEUE WITH PREEMPTIVE PRIORITY 5. INTRODUCTION In last chater we discussed the case of non-reemtive riority. Now we tae the case of reemtive riority. Preemtive riority means, the service to the ongoing job will be reemted by the new arrival of higher riority. If the riority disciline is reemtive resume, then service to the interruted job, when it restarts, continues from the oint at which the service was interruted. For the reemtive non resume case, service already rovided to the interruted job is forgotten and its service is started again from the beginning. Note that there may be loss of wor in the reemtive non-resume riority case. Such loss of wor will not haen in the case of the other two riorities. Since the service times are assumed to be exonentially distributed, they will satisfy the memoryless roerty and that, therefore, the results will be the same both for the reemtive resume and reemtive non-resume cases. We consider here queues with reemtive riority discilines. There is quite some literature on single server riority queueing systems. However, multi-server queueing systems have received much less attention. Such models have been studied by Mitrani et al. [65], Gali et al. [3,3], Kao et al. [ 46,47], Kella and Yechiali [49] and Wagner [89,9,9]. 5. ASSUMPTIONS In this chater, we consider an M/M//K queueing system with riority. The assumtions of the system model are as follows:. The system at any instant of time is one of the two modes of oeration.. All distribution are negative exonential, i.e, all transition rates are constants.

2 3. The robability of two or more random events occurring simultaneously is negligible. 5.3 NOTATIONS i mode of system oeration, i, N maximum number of customers in the system n current number of customers in the system i arrival rate in mode i µ i service rate in mode i L mean number in the system W mean waiting time in the system L q mean number in the queue W q mean waiting time in the queue 5.4 SYSTEM MODEL We first consider the - riority M/M/ queue with infinite buffers oerating with a reemtive riority disciline.this queue may be analysed by drawing a state transition diagram Fig. 5.. Figure 5.: State Transition Diagram for -Priority M/M/ Queue with Preemtive Priority Since all the flows are nown, the equilibrium robabilities of each of the states may be found by writing and solving the corresonding balance equations with the normalisation condition. As an 3

3 4,,,,,,,,,, ) ( ) ( ) (.,,,, examle, consider the - riority M/M// queue with reemtive riority. State transition diagram for M/M// queue with -riority is shown in Fig. 5..,,,, Figure 5.: State Transition Diagram for -Priority M/M// Queue with Preemtive Priority 5.5 STEADY- STATE PROBABILITY In this art, we derive the steady-state robabilities by the Marov rocess method. Let P(n) be the robability that there are n customers in the system. Alying the Marov rocess theory, we obtain the following set of steady-state equations. Matrix form of these equations is:

4 5.,,,,,,,,,..,,,, It is a homogenous system of equation. This system has infinite no. of solutions. On solving these equation we get; Using we get, Therefore,

5 5.6 NUMERICAL RESULTS In this subart, we resent some numerical examles to demonstrate how the various arameters of the model influence the erformance measures of the system. We fix the maximum number of customers in the system N. Table 5.: Performance measures for /µ.8 /µ L L q W W q First, we select the /µ.8 and change values of arrival rate of customer /µ.the numerical results are summarized in Table. This shows that: (i) the length of queueing system increases with the increasing /µ.(ii) The length of the queue increases uto /µ from. to.3, it decrease at.4, again increases from.5 to.6, again decrease at.7 and then increases. (iii) The Waiting time in queueing system decreases with increasing /µ. (iv) The Waiting time in queue again decreases with the increasing /µ. The following grah shows the erformance of the model at /µ.8. 6

6 L /µ Figure5.3: Grah between /µ and length of system Lq /µ Figure 5.4: Grah between /µ and queue length 7

7 4.5 W /µ Figure 5.5: Grah between /µ and waiting time of system.5 Wq /µ Figure 5.6: Grah between /µ and waiting time of queue 8

8 5.7 CONCLUSION: In this chater, we considered an M/M// queueing system with riority. We develoed the equations of the steady state robabilities and derived the matrix form solution of the steady-state robabilities. We also gave some erformance measures of the system. Some numerical examles were resented to demonstrate how the various arameters of the model influence the behavior of the system. 9