Applied Mathematical Sciences, Vol. 8, 2014, no. 92, 4585-4592 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.45388 Transient Solution of a Multi-Server Queue with Catastrophes and Impatient Customers when System is Down G. Arul Freeda Vinodhini Saveetha School of Engineering Saveetha University, Chennai-32, India V. Vidhya Saveetha School of Engineering Saveetha University, Chennai-32, India Copyright 2014 G. Arul Freeda Vinodhini and V. Vidhya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We consider an M/M/c queuing system, which occasionally suffers disastrous failure and all the customers are lost. The repair mechanism starts immediately. When the system is down, the stream of arrivals continue. However the new arrivals become impatient and activate their own timer. If the system is still down when the timer expires, the customer abandons the system never to return. We obtain the exact transient solution for the system by a direct approach. Mathematics Subject Classification: Primary 60K25, Secondary 68M20 Keywords: M/M/c Disaster Impatience - Transient Probabilities Generating function
4586 G. Arul Freeda Vinodhini and V. Vidhya 1 Introduction In real life, many queuing situations arise which are not reliable and disasters may occur leading to loss of several or all customers. Such situations are common in computer network applications, telecommunication applications that depend on satellites and inventory system that store perishable goods. Customer impatience is also observed in queue models like impatient telephone switch board customers, packet transmission etc Also many traditional studies analyze queuing system in steady state, requiring appropriate warm up period. However, in many cases the system being modeled never reaches steady state and hence do not accurately portray the system behavior, as in military air traffic control, emergency medical service etc The earlier works on the transient behavior of queues in literature were published in the late 1950 s and in early 1960 s. The transient solution of various single server queue models like state dependent queues [4], potential customers discouraged by queue length[3], feedback with catastrophes [8] etc are studied in the literature. In 2010, R. Sudhesh derived transient solution of a single server queue with system disaster and customer impatience [7]. In 1989, P.R. Parthasarathy and M. Sharfli gave the transient solution to many server Poisson queue [2].R.O. Al Seedy et al derived the transient solution of M/M/c queue with balking and reneging [1]. Transient solution of a multi-server Poisson queue with N policy was derived by P.R. Parthasarathy and R. Sudhesh [5]. In the present paper, multi-server queue is considered in the presence of system disaster and customer impatience when the system is down. A closed solution for transient probabilities of the model is derived. This work expands the previous work by Yechiali [9] where the steady state probabilities of the model are derived. It also uses the results obtained by Sudhesh [7].The rest of the paper is organized as follows. Section 2 explains the governing equations of the model. In Section 3 the transient probabilities of the model are derived. Section 4 presents the validity of the transient probabilities obtained. Section 5 is the conclusion and finally the references are given. 2 Model Description 2.1 The model and transition diagram Customers arrive at an M/M/c type queue according to a Poisson process with rate. Service times are exponentially distributed with mean 1. The system suffers disastrous breakdowns, occurring when the server is at functioning phase, at a Poisson rate. When the system fails all servers stop working and all customers present are rejected and lost. Upon failure, a repair process starts imme-
Transient solution of a multi-server queue 4587 diately with exponentially distributed with mean 1. Impatient customers activate a timer exponentially distributed with mean 1, such that if the repair process has not been completed by the time expires, the customers abandons the system never return. The above process generates a two dimensional continuous time Markov process as follows. Let J indicate the system s phase: J=1 denote that the system is functioning and serving customers, while J=0 indicates the system is down, undergoing a repair process. The system alternates between the two phases and it turns out that the proposition of time the system stays in the two phases are not affected by the number of servers of the system. If L denotes the number of customers in the system, then (J, L) define a two dimensional continuous Markov chain with state space, : 0, 1; 0, 1, 2 2.2 Governing equations Let denote the system state probabilities in transient state. From the transition diagram the Kolmogorov differential difference equations are given by, (2.1), 1,, 1 (2.2) (2.3), 1,, 1 1 (2.4),,, (2.5) Also, assume that the system is working and there are no customers in the system at time 0. (i.e.) 0 1 and 0 0 1. Clearly, for any 0, 1. 3 Transient Solution 3.1 Expressions for The equations for the phase J=0 (2.1 and 2.2) are same as solved by R. Sudhesh [10]. 3.2 Expressions for 3.2.1 Evaluation of for n = c, c+1, c+2 If we define,,, with,, and, 0 1,then equations 2.3, 2.4 and 2.5 reduce to the differential equation.,, 1,,,,
4588 G. Arul Freeda Vinodhini and V. Vidhya Solving the above equation, we get, 1,, Taking 2 and 1,,, (2.6) we have Equating coefficients of on both sides we get,,,, +,,,,,,, (2.7) Put n = 0 on both sides we get,,,, +,,,,,, (2.8) Put in (2.7) and using.. 0,, +,,,,
Transient solution of a multi-server queue 4589,, (2.9) From (2.7) and (2.9) for n =1, 2, 3 we get,, =,,,,, (2.10) 3.2.2: Evaluation of for n = 0, 1, 2 (c-1) Consider the system of equations 2.1 and 2.2 along with the condition 2.8. Equations 2.1 and 2.2 can be expressed in the form 1, (2.11) where,,,,,,,, 0 0 0 1 and 1 1 1 1 1, 0, 1, 2, 2, 0, 1, 2, 2 1 1, 0, 1, 2, 3 Taking Laplace transform of 2.11, 1, 0 (2.12) Also,,, (2.13) Thus only,has to be found. Combining equations 2.8, 2.12 and 2.13 we get,, (2.14), where,. (2.15) To find: For smaller order matrices can be found by the usual procedure. For higher order matrices, we follow the procedure given by Raju and Bhat in [6]. Let where
4590 G. Arul Freeda Vinodhini and V. Vidhya,,,,, 0,1,2 3 (2.16), 2, For 0, 1, 2 2 where, are recursively given as, 1 0,1,2 2 ;, 0,1,2 3,,,,, 0,1,2 3 2,, 0,1,2 3 2 2 and, 0 for other values of k and j. Using above result in (2.14),,,,,, (2.17) and from (2.12) for 0, 1, 2 2, 1,,,,, (2.18) We observe that, are all rational algebraic functions in s. The cofactor of the (k, j)th element of is a polynomial of degree 2. In particular, the cofactors of the diagonal elements are polynomials in s of degree 2 with the leading coefficient equal to 1.In fact, 0 is the characteristic equation of A. Since, 0 the characteristic roots of A are all distinct and negative [10]. Hence the inverse transform, of, can be obtained by partial fraction decomposition. Let, 0, 1, 2 2 be the characteristic roots of the matrix. Then,, where, lim.s,,. (2.19) Defining 1 1 0,1,2 2 (2.20) Where can be resolved as, with lim 1 Thus the inverse transform of is given by, j 0, 1 2. Using (2.20) in (2.17),
Transient solution of a multi-server queue 4591,, which simplifies to, 1 1, On inversion we get,, 1, (2.21) where can be obtained by inverting (2.15) as,, 2, Inverting (2.18), for k = 0, 1,2 c-2 we get, 1,,,,, (2.22) Thus the transient state probabilities for the state J=1 are given by the equations (2.10), (2.21) and (2.22). 4 Deductions (i) When c=1(assuming m=0 for initially there are zero customers), the problem reduces to the model derived in [7]. (ii) When and tends to 0(assuming 0for initially there are zero customers), the model becomes an ordinary transient multi server model and hence the transient probabilities coincide with [2]. 5 Conclusion Transient solution for multi server queue with impatience and disaster is derived. Such situations may occur when a computerized information center suddenly becomes inoperative, in some real life public service systems when potential customers leave the system one by one when the system is down. For handling such problems transient solutions portray the system better than a steady state solution.
4592 G. Arul Freeda Vinodhini and V. Vidhya References [1] Al-Seedy R.O., A.A. El. Sherbiny, S.A.ElShehawy, S.I. Ammar, Transient solution of M/M/c queue with balking and reneging, Computers and Mathematics with Applications 57(2009), 1280 1285. [2] Parthasarathy P.R, M. Sharafali, Transient solution to the many server Poisson queue: A simple approach, Journal of applied probability 26(1989), 584-594. [3] Parthasarathy P.R., Selvaraju. N, Transient analysis of a queue where potential customers are discouraged by queue length, Math. Probl. Eng.7(2001), 433-454. [4] Parthasarathy P.R., Sudhesh.R, Exact transient solution of state dependent birth death processes. J.appl.Math.Stch. Anal.2006,(2006) Article ID 97073, 1-16. [5] Parthasarathy P.R., Sudhesh.R, Transient solution of a multi-server Poisson queue with N Policy, Computers and Mathematics with Applications 55(2008), 550-562. [6] Raju S. N., U.N. Bhat, A computationally oriented analysis of the G/M/1 queue, Opsearch 19(1982), 67-83. [7] Sudhesh. R, Transient analysis of a queue with system disasters and customer impatience, Queueing Syst 66(2010), 95-105. [8] Thangaraj. V. Vanitha. S, On the analysis of M/M/1 feedback queue with catastrophes using continued fractions, International journal of Pure and Applied Mathematics 53(2009), 133-151. [9] Uri Yechiali, Queues with system disasters and impatient customers when the system is down, Queueing Syst 56(2007), 195-202. [10] W. Lederman, G.E.H Reuter, Spectral theory for the differential equations of simple birth and death processes, Philosophical Transactions of the Royal society London 246(1954), 321-369. Received: June 5, 2014