A Balked M/D/1 Feedback Queueing Models

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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: X.Volume2,Issue Ver. III (Jan.-Feb.206)PP A Balke M/D/ Feeback Queueing Moels Dr. V. K. Gupta, Tabi Nanan Joshi 2, Dr. S. K. Tiwari (Govt. Mahav Science College, Vikram University, Ujjain, Inia) 2 (School of Stuies in Mathematics, Vikram University, Ujjain, Inia) (School of Stuies in Mathematics, Vikram University, Ujjain, Inia) Abstract: In this paper, we present an analysis for an M/D/ queueing system with balking. It is assume that arriving customers may balk with a probability accoring to an exponential istribution. We evelop the equations of the steay state probabilities an solution of the steay-state probabilities. The steay-state solution of the moels is erive. Some useful measures of performance are erive. Some queueing moels are obtaine as particular cases of the moel are iscusse. Keywors: Balking; Reneging; Steay-state Solution. I. Introuction In real life, many queueing situations arise in which there may be a tenency for customers to be iscourage by a long queue. As a result, the customers either ecie not to join the queue (i.e. balk) or epart after joining the queue without getting service ue to impatience (i.e. renege). A queue, or a waiting line, involves arriving items that wait to be serve at the facility which provies the service they seek. Queuing theory is concerne with the statistical escription of the behavior of the queues with result. Balking an reneging are not only common phenomena in queues arising in aily activities, but also in various machine repair moels. For relate literature may refer to [7], [], an references therein. An attractive example of the occurrence of balking an reneging is given in Ancker an Gafarian [2]. Queueing systems with balking, reneging, or both have been stuie by many researchers. Haight [] first consiere an M/M/ queue with balking. An M/M/ queue with customers reneging was also propose by Haight [4]. In this paper we have iscusse about a steay state solution of the orere queuing problem with reneging. A customer may enter the queue, but after a time lose patience an ecie to leave. In this case the customer is sai to have renege. A unit reneges (i.e., becomes impatient an leaves without having been serve) after joining the queue if it is ecie that the wait will be longer than can be tolerate. Here the waiting line is of Poisson balking probability which epen not only on the number of customers in the system, but also on the rate of service in the system. A queuing situation with the following characteristics has been consiere. A customer receives the service immeiately, when the system is empty. But a joining customer that has to wait for service ue to impatience may leave i.e., the customer may renege. Service is performe on the customer at the hea of the line. One service has commence on a customer, it remains until the completion of service. Barrer [0] has stuie the problem of a unit leaving a queue after having waite longer than an acceptable time. O. Brien [2] has also foun the solution of some queuing problem.miller [22] an Konigsberg [] have also stuie about balk queue an queuing with special service. In this paper we have attempte to fin out a steay state solution of queue, when the Poisson probabilities epen not only on the number of the customers in the system. Customers are the backbone of any business, because without customers there will be no reason for a business to operate. Customer impatience leas to loss of potential customers. It has become a highly challenging problem in the current era of cut-throat competition. Queueing with customer impatience has special significance for the business worl as it has a very negative effect on the revenue generation of a firm. Therefore, the concept of customer retention assumes a tremenous importance for the business management. Customer retention is the key issue in the organizations facing the problem of customer impatience. Firms are employing a number of customer retention strategies to maintain their businesses. An impatient customer (ue to reneging) may be convince to stay in service system for his service by utilizing certain persuasive mechanisms. Such customers are terme as retaine customers. When a customer gets impatient, he may leave the queue with some probability, say p 2 an may remain in the queue for service with some complementary probability q 2 (= p 2 ). Recently, Kumar an Sharma [20] stuy the retention of renege customers in an M/M//N queuing moel an perform sensitivity analysis of the moel. Kumar an Sharma [2] stuy M/M//N queuing system with retention of renege customers an DOI: / Page

2 balking. They exten the work of Kumar an Sharma [20] by taking balking aspects in their moel to stuy the effect of probability of retaining the renege customers on expecte system size. They perform the sensitivity analysis of the moel. We assume that after the completion of service, each customer may rejoin the system as a feeback customer for receiving another regular service with probability p an may not join with complementary probability - p. Sharma an Kumar [25] stuy single-server finite capacity, Markovian queue with feeback an retention of renege customers. They perform steay-state analysis of the moel. Sharma an Kumar [25] further stuy M/M//N feeback queuing moel with balking an retention of renege customers in the same year. They obtain steay-state solution of the moel. They erive important performance measures of the moel. Some queueing moels are erive as special cases of the moel. This paper iscusse the queueing moel for eterministic service time. Some useful measures of performances are erive. Some queuing moels are obtaine as particular cases of the moel. literature survey is also presente. Here, also present the steay-state analysis of a single-server Markovian feeback queuing system with retention of renege customers. M/D/ queuing moel with retention of renege customers consiere. In the case of balking, immeiately on arrival a unit ecies not to join the queue, perhaps because of the length of the queue or because of other information on the length of service. II. Literature Survey The earlier work on feeback is foun in Takacs [26]. He stuies queue with feeback to etermine the stationary process for the queue size an the first two moments of the istribution function of the total time spent in the system by a customer. Davignon an Disney [8] stuy single server queues with state epenent feeback. Santhakumaran an Thangaraj [24] consier a single server feeback queue with impatient an feeback customers. They stuy M/M/ queueing moel for queue length at arrival an obtain result for stationary istribution, mean an variance of queue length. The combine effects of balking an reneging in an M/M//N queue have been investigate by Ancker an Gafarian []. Wang an Chang [8] extene this work to stuy an M/M/c/N queue with balking, reneging an server breakowns. Thangaraj an Vanitha [27] obtain transient solution of M/M/ feeback queue with catastrophes using continue fractions. The steay-state solution, moments uner steay state an busy perio analysis are calculate. Ayyappan [4] stuy M/M/ retrial queueing system with loss an feeback uner non-pre-emptive priority service by matrix geometric metho. Customer retention in a single-server, finite capacity, Markovian queueing moel is stuie by Kumar an Sharma [20], the relate work on customer retention an its application is in [25]. Several excellent surveys on these vacation moels have been one by Doshi [9] an Takagi [5]. However, there are only a few works that take into consieration balking an reneging phenomena involving server vacations. The reaers may refer to Zhang et al. [28] where an M/M//N queueing system with balking, reneging an server vacations was consiere. The concept of customer impatience appears in queueing theory in the work of Haight []. Haight [4] stuies queuing with reneging. Ancker an Gafarian [] stuy M/M//N queuing system with balking an reneging an erive its steay-state solution. Al-seey et al. [5] stuy M/M/c queue with balking an reneging an erive its transient solution by using the probability generating function technique an the properties of Bessel function. Chouhury an Mehi [6] stuy customer impatience in multi-server queues. They consier both balking an reneging as functions of system state by taking into consieration the situations where the customer is aware of its position in the system. Kapoistria [9] stuy a single server Markovian queue with impatient customers an consier the situations where customers abanon the system simultaneously. He consiers two abanonment scenarios. In the first one, all present customers become impatient an performe synchronize abanonments, while in the secon scenario; the customer in service is exclue from the abanonment proceure. He extens this analysis to the M/M/c queue uner the secon abanonment scenario also. In this paper, infinite capacity, single-server eterministic service feeback queueing moel with customer retention an balking is stuie. The steay-state analysis is performe. The steay state probabilities are obtaine iteratively in case of reneging an retention of renege customers. The performance measures are erive an some queuing moels are iscusse as particular cases of the moel. III. A Balke M/D/ Feeback Queuing Moel It is necessary to know the reaction of the customer while entering in the system. A customer may ecie to wait no matter how long the queue becomes, or there may be a case, if the queue is too long, the customer may ecie to not to enter in the system. If a customer ecies not to enter the queue upon arrival, then the customer is sai to have balke. In case of feeback, after the completion of service, each customer may rejoin the system as a feeback customer for receiving another regular service with probability p an may not join with complementary probability - p. We consier an M/D/ feeback queuing moel with reneging. The DOI: / Page

3 reneging times are assume to exponentially istribute with parameter ξ. It is visualize that a renege customer may be convince by applying certain convincing metho to stay in the system for his service. Thus, there is a probability say, q 2 that a renege customer may be retaine in the system an may not be retaine with some complementary probability say, p 2 =( q 2 ). The moel in this section is extension of the moel M/D/ Feeback Queueing Moels with Retention of Renege Customers by incluing the aspect of balking in the system. On arrival, the customer may balk with certain balking probability. Let the probability of balking is q or the customer may join the queue with probability p such that p +q =. The ifferential-ifference equations of the moel are: t t = t + μq P (t) Where P n (t) is the probability of n in the system at time t. Here, we use eterministic service time moel. The approach we use is similar to to be foun in saaty [2] an which is essentially originally ue to Crommelin [7]. In the M/D/ queueing moel the arrival rate is an the constant service time (say b=/µ). Now we rescale our parameter as = /b an µ=[6] so that the traffic intensity ρ = µ The ifferential-ifference equation of the moel accoring to above conition becomes: t t = b t + q P (t)... () t P t = + q P t + q +ξp 2 P 2 t + t. (2) remains unaffecte. t P n t = + q + n ξp 2 P n t + q +nξp 2 P n + t + P t n. () In steay state, lim t P n an therefore P t n t = 0 as t an hence the equation (7) to (9) gives the ifference equations as 0 = b t + q P (t).(4) 0 = + q P t + q +ξp 2 P 2 t + t. (5) 0 = + q + n ξp 2 P n t + q +nξp 2 P n+ t + P t On solving iteratively, the steay state probabilities of system size are given by n 2. (6) b p ; n (7) With = + + n =2 q+kξp 2.(8) The steay state probabilities exist if + + < 4. The Expecte System size, L s IV. Measures Of Performance: L s = np n DOI: / Page

4 L s = n 4.2 The Expecte Queue Length L q L q = L q = n np n 4. The Expecte waiting time W s W s = b n 4.4 The Expecte waiting time in the queue W q W q = b n q 4.5 The Expecte number of customers serve, (E cs ) The expecte number of customer serve is given by 4.6 Rate of Abanonment, R aban E cs = q E cs = q P n as µ = R aban R aban = b q = b P n E cs 4.7 Expecte number of waiting customers actually waits, E cw (n )P n E cw = E cw = (n ) P n Where is given in eq. (8). 4.8 PARTICULAR CASES OF THE MODEL : Here we introuce M/D/ feeback queueing moel with retention of renege customer an balking. In the absence of retention of renege customer an balking. When the probability of retention of Renege customer is zero. i.e., q 2 =0. We get b p q + kξ ; n DOI: / Page

5 With = + + N n =2 q+ kξ When the capacity of the system is finite i.e., N. With = b p + + ; N n =2 n N q+kξp 2 In the absence of balking, when p = the moel reuce to M/D/ feeback queueing moel with retention of renege customers. V. Conclusion This theory has been evelope in an attempt to preict the fluctuating emans of queue length an units in the system an to enable an enterprise to provie aequate service for its customers an poisson balking probabilities. In this paper, we have stuie customer retention in an M/D/ feeback queuing moel reneging an balking. The steay-state solution is obtaine for the queuing moel. Some important measures of performance are erive. Some important queuing moels are obtaine as particular cases of the moel. References []. Ancker Jr., C. J. an Gafarian, A. V., Some Queuing Problems with Balking an Reneging. I, Operations Research, vol., no., pp , 96. [2]. Ancker Jr., C. an Gafarian, A. Queueing with impatient customers who leave at ranom. Journal of Inustry Engineering, XIII, 84 90, 962. []. A. Shawky. The machine interference moel: M/M/C/K/N with balking, reneging an spares. Opsearch, 7, 25 5, [4]. Ayyapan, G., Muthu Ganapathi Subramanian, A. an Sekar, G., M/M/ Retrial Queueing System with Loss an Feeback uner Non-pre-emptive Priority Service by Matrix Geometric Metho, Applie Mathematical Sciences, vol. 4, no. 48, pp , 200. [5]. Al-Seey, R.O., El-sherbiny, A.A., El-Shehawy, S.A, an Ammar, S.I., Transient Solution of the M/M/c queue with balking an reneging, Computer an Mathematics with Applications, vol. 57, no. 8, pp , [6]. Chouhury, A., an Mehi, P., Balking an reneging in multiserver Markovian queuing systems, International Journal of Mathematics in Operational Research, vol., no. 4, 77-94, 20. [7]. Crommelin, C. D. 92. Delay Probability formulae when the holing times are constant. P. O. Electrical Engineering Journal 25, [8]. D' Avignon, G.R. an Disney, R.L., Single Server Queue with State Depenent Feeback, INFOR, vol. 4, pp. 7-85, 976. [9]. Doshi, B. Single server queues with vacation: a survey. Queueing System, (), 29 66, 986. [0]. D.Y.Barrer,Queueing with impatient customers an inifferent clerks, Operation Research,Vol.5,No []. E.Koenigsberg,Queueing with special service, Operation Research,Vol.4,2-220, 956. [2]. G.G.O,Brien, The solution of some queuing problems, J.Soc.In.Appl.Math., Vol.2,-42,954. []. Haight, F. A., Queuing with balking, I, Biometrika, vol. 44, pp , 957. [4]. Haight, F. A., Queueing with Reneging, Mettrika vol. 2, pp , 959. [5]. H. Takagi. Queueing Analysis, A Founation of Performance Evaluation, Volume: Vacation an Priority Systems, Elsevier, 99. [6]. Gross, D. Shortle, J. Thompson, J. an Harris, C., Funamentals of Queueing Theory, Forth eition, Wiley, Poisson Input, Constant Service. pp [7]. J-C. Ke an K-H. Wang. The reliability analysis of balking an reneging in a repairable system with warm stanbys. Quality an reliability Engineering International, 8, 2 28, [8]. K-H. Wang an Y-C. Chang. Cost analysis of a finite M/M/R queueing system with balking, reneging an server breakowns. Mathematical Methos of Operations Research, 56(2), 69 80, [9]. Kapoistria, S., The M/M/ queue with synchronize abanonments, Queuing Systems, vol. 68,pp , 20. [20]. Kumar, R. an Sharma, S.K., M/M//N queuing system with retention of renege customers, Pakistan Journal of Statistics an Operation Research, vol. 8, no. 4, pp , 202. [2]. Kumar, R. an Sharma, S.K., M/M//N queuing system with retention of renege customers an balking, American Journal of Operational Research, vol. 2, no., pp. -5, 202. [22]. R.G.Miller,A contribution to the theory of balk queues,j.roy Statist.Soc., Vol.2,No.2,20-7,959. [2]. Saaty, T. L. 96, Elements of Queueong Theory with Applications McGraw Hill, New York. [24]. Santhakumaran, A. an Thangaraj, V., A Single Server Queue with Impatient an Feeback Customers, Information an Management Science, vol., no., pp. 7-79, [25]. Sharma S.K. an Kumar R., A Markovian feeback Queue with Retention of Renege Customers an Balking, AMO-Avance Moelling an Optimization, vol. 4, no., pp , 202. [26]. Takacs, L., A Single Server Queue with Feeback, The Bell System Tech. Journal, vol. 42, pp. 4-49, 96. [27]. Thangaraj, V. an Vanitha, S., On the Analysis of M/M/ Feeback Queue with Catastrophes using Continue Fractions, International Journal of Pure an Applie Mathematics, vol. 5, no., pp. -5, 2009 [28]. Y. Zhang, D. Yue an W. Yue. Analysis of an M/M//N queue with balking, reneging an server vacations. Proc. of the Fifth International Symposium, 7 47, DOI: / Page