A New Measure for M/M/1 Queueing System with Non Preemptive Service Priorities

Transcription

1 International Journal of IT, Engineering an Applie Sciences Research (IJIEASR) ISSN: Volume, No., October A New Measure for M/M/ Queueing System with Non Preemptive Service Priorities Inu Jinal, Shweta Sharma 2 Post Grauate Govt. College for Girls, Sector 42, Chanigarh 2 Rayat Bahra College of Engineering an Bio-Technology for Women, Mohali (Punjab) ABSTRACT This paper stuies M/M/ queue uner non preemptive service priority iscipline. The unit with higher priority gets the service first without interrupting the service of non- priority unit if it is alreay in service. The stuy obtains the Laplace Transform of explicit time-epenent probabilities of exactly i arrivals, j epartures of priority units an k arrivals, l epartures of non-priority units by time t by solving the Laplace Transform of ifference equations iteratively. By inverting the Laplace Transforms actual probabilities can be known. Few explicit results have been erive which helpe in the verification of the moel. With the help of these explicit transient state probabilities, other measures of interest can be obtaine. Keywors Queues, Priority, Transient behaviour. INTRODUCTION It was Cobham [3] who consiere non-preemptive priority queueing iscipline with Poisson input an exponential holing time. Phipps [4] extene the stuy to the case when the numbers of priorities are continuous. White an Christie [8] obtaine the steay state solution of preemptive priority queues with Poisson arrivals an exponential service time. Heathcote [6] foun the time-epenent solution of the same moel. Later Miller [2], Heathcote [7, 8], Jaiswal [9, 0], Gaver [4], Hawkes [5], Shara [5-7], Kao & Wilson [], Choi et.al [2], Balter et.al [] stuie some queueing systems with priority as queue iscipline. In the present paper, the concept of Pegen an Rosenshine [3] is applie to fin out the explicit time epenent probabilities for exact number of arrivals an epartures of priority units as well as of non-priority units by a given time recursively. The practical situation which correspons the above problem can be that of a repair shop of electronic items. In winter season, repair of electric heaters will be on priority as compare to other electronic items. The mechanic can know the number of repaire heaters an number of total heaters receive for repair an similarly for other electronic items by a given time. QUEUEING MODEL The queueing system stuie in this paper is escribe by the following assumptions: (i) The priority units an non-priority units arrive in a Poisson istribution with parameters λ an λ 2 respectively. (ii) The service time follows an exponential istribution with parameter µ. (iii) When there is one priority unit in the system then it may or may not be in channel. Let p an q be the probabilities that the service is being one of priority or non priority units such that pq. (iv) When there are more than one priority units in the system then service will be one of priority unit only. (v) The stochastic processes involve viz. (a) Arrival of units. (b) Departure of units; are statistically inepenent.

2 International Journal of IT, Engineering an Applie Sciences Research (IJIEASR) ISSN: Volume, No., October Define P i,j,k,l t Probability of i arrival, j eparture of priority units an k arrival, l eparture of non - priority units by time t. i j, k l P n t Probability of n arrivals by time t. i k P i,j,k,lt, where i k n j 0 l 0 Initially P 0,0,0,0,F 0 P 0,0,0,0,B t0 The ifference- ifferential equations governing the system are: t P,,, t µp,,, t P,,, t P,,, t µp,,, t j i 2,0 l k () t P,,,t µp,,, t P,,, t µpp,,, t µp,,, t i, l k (2) t P,,,t µp,,, t P,,, t P,,, t µp,,, t µqp,,, j i 2,0 l k (3) t P,,,t µp,,, t P,,, t P,,, t i 2,0 l k (4) t P,,, t µp,,, t P,,, t P,,, t µqp,,, t l k (5) t P,,, t µp,,, t P,,, t µp,,, t l k 6 t P,,,t µp,,, t P,,, t µp,,, t j i 2, k t P,,,t µp,,, t P,,, tµp,,, t 7 µqp,,, t i 2,k (8) t P,,,t P,,, tµp,,, t µqp,,, t i,k 9 t P,,, t µp,,, t P,,, t µpp,,, t i,k 0 t P,,, t µp,,, t P,,, t i 2, k t P,,, t P,,, t µp,,, t t P,,,t µp,,, t k 2 P,,, t P,,, t µqp,,, t i 2,k (3)

3 International Journal of IT, Engineering an Applie Sciences Research (IJIEASR) ISSN: Volume, No., October t P,,, t µp,,, t P,,, t k 4 t P,,, t P,,, t µp,,, t t P,,, t µp,,, t P,,, t 5 µp,,, t i 2 6 t P,,, t µp,,, t P,,, t µqp,,, t k 7 t P,,, t µp,,, t P,,, t P,,, t k 8 t P,,, t µp,,, t P,,, t 9 t P,,, t P,,, t 20 Taking Laplace transform of P,,, t given by e P,,, t t, where Re s > 0, equations () to (20) transforms to equations (2) to (40). µ P,,, s P,,, s µp,,, s j i 2,0 l k (2) i µ µp,,, s P,,, sµpp,,, s µp,,, s i, l k (22) P,,, s P,,, sµp,,, s µqp,,, s i 2, l k (23) µ P,,, s P,,, s i 2,0 l k (24) µ P,,, s µqp,,, s l k (25) µ P,,, s µp,,, s l k (26) µ P,,, s µp,,, s j i 2, k (27) µp,,, s P,,, s µp,,, sµqp,,, s µ µp,,, s i 2,k (28) µp,,, s i,k (29) µ P,,, s µp,,, s µp,,, s i 2, k (30) k (3) P,,, s P,,, s µp,,, s

4 International Journal of IT, Engineering an Applie Sciences Research (IJIEASR) ISSN: Volume, No., October i 2,k (32) µ P,,, s µpp,,, s i,k (33) µ P,,, s k (34) µp,,, s i (35) µp,,, s P,,, s µp,,, s i 2 (36) µ µqp,,, s k (37) µ P,,, s k (38) µ (39) (40) Solving (2) to (40) iteratively Laplace transform of all the probabilities P,,, t for all values of i, j, k an l can be known an by using inverse Laplace transform P,,, t can be completely known. FEW EXPLICIT RESULTS µ µ µ µ µ µ µ µ 2 µ µ µ 2p µ µ µ 2q µ 2 µ µ µ µ µ µ µ µ µ

5 International Journal of IT, Engineering an Applie Sciences Research (IJIEASR) ISSN: Volume, No., October µ µ µ µ µ µ Hence, the verification. This paper has analyse M/M/ queue with nonpreemptive priorities with a single server, Poisson arrivals of priority an non-priority units with ifferent parameter. Service times are exponentially istribute. We have erive general equilibrium equations for i arrivals, j epartures of priority units an k arrivals, l epartures of non priority units. Future work will focus on solving the system of equations by writing the algorithm for above queueing system an implementing this algorithm in MATLAB. VERIFICATION OF THE MODEL Taking Laplace Transform of P t given by P s e P tt an from explicit results P s P s P s an so on. By taking Laplace inverse transform, we get P te P t t e! P t t e 2! an so on. Therefore, P t e!! REFERENCES [] Balter M. H., Osogami T., Wolf A.S. an Wierman A., (2005) Multi-Server Queueing Systems with Multiple Priority Classes. Queueing Systems: Theory an Applications [2] Choi B.D., Kim B., an Chung J., (200) M/M/ Queue with Impatient Customers of Higher Priority. Queueing Systems: Theory an Applications [3] Cobham A.,(954) Priority Assignment in Waiting Line Problems. Operations Research [4] Gaver D.P., (962) A waiting line with interrupte service incluing priorities. J. Roy. Stat. Soc. B [5] Hawkes A.C., (965) The time epenent solution of a priority queue with preemptive priorities. Operations Research [6] Heathcote C.R., (959) The time epenent problem for a queue with preemptive priorities. Operations Research [7] Heathcote C.R., (960) A Simple queue with several priority classes. Operations Research [8] Heathcote C.R., (96) Preemptive Priority Queueing. Biometerika [9] Jaiswal N. K., (96a) Preemptive resume priority queue. Operations Research e e

6 International Journal of IT, Engineering an Applie Sciences Research (IJIEASR) ISSN: Volume, No., October [0] Jaiswal N. K., (962) Time epenent solution of the hea of the line priority queue. Jr. Roy Statistic Soc. B [] Kao E. an Wilson S., (999) Analysis of Nonpreemptive Priority Queues with Multiple Servers an Two Priority Classes. European Journal of Operational Research [2] Miller R.G., (960) Priority Queues. Ann. Math. Stat [3] Pegen C. D. an Rosenshine M., (982) Some New Results for the M/M/ queue. Management Science [4] Phipps T.E., (956) Machine Repair as Priority Waiting Line Problem. Operations Research [5] Shara, (973a) On a Certain Type of Priority Queueing Problems. Metrika [6] Shara, (979) A Priority Queueing Problem with Intermittently Available Phase Type Service, Cahiers u CERO. 2 [7] Shara, (98-983) Preemptive Resume Priority Queueing Problem with Batch Arrivals an Departures an Intermittently Available Server. Journal Of Mathematical Sciences [8] White H. an Christie L.S., (958) Queueing with Preemptive Priorities or with Breakown. Operations Research