Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes

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1 Information Management Sciences Volume 18, Number 1, pp. 63-8, 27 Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes B. Krishna Kumar S. Pavai Madheswari Anna University India Anna University India K. S. Venkatakrishnan Anna University India Abstract This paper presents a transient solution for the system size in an M/M/2 queue where the service rates of the servers are not identical with the possibility of catastrophes at the system. The time dependent probabilities for the number in the system are obtained. The steady state probabilities of the system size are also provided. Some important performance measures are derived. Keywords: System Size, Heterogeneous Servers, Catastrophes, Steady State Probability. 1. Introduction Multi-sever queueing systems arise in congestion problems of telephone systems computer networks. Computer systems or data transmission networks deal with systems having multiple resources ( Central processing units, Channels, Memories etc.). transaction that cannot immediately get hold of the required resources is usually queued up in a buffer until the resource becomes available. This characteristic makes the computer systems amenable for analysis using multi-server queueing models. A complete description of situations with such queueing analysis of computer systems can be found in Lavenberg [1]. Heterogeneity of service is a common feature of many real multi-server queueing situations. The heterogeneous service mechanisms are invaluable scheduling methods that allow customers to receive different quality of service. Heterogeneous service is clearly a Received December 25; Revised February 26; Accepted June 26. Supported by ours. Any

2 64 Information Management Sciences, Vol. 18, No. 1, March, 27 main feature of the operation of almost any manufacturing system. The role of quality service performance are crucial aspects in customer perceptions firms must dedicate special attention to them when designing implementing their operations. For this reason, the queues with heterogeneous servers have received considerable attention in the literature. Surprisingly, heterogeneous machine centers were only rarely treated in research specially in queueing theory. A Markovian queueing system with balking two heterogeneous servers has been discussed by Singh [15]. A control model for a machine center with two heterogeneous servers has been introduced by Liu Kumar [12]. Some attention has also been given to multi-server queueing systems with different service time distributions for different servers (Lazowaska et al. [11]). Recently, the queueing systems with heterogeneous servers have been considered by Mittler [13] Dörrsam [5], to study the impact of heterogeneity of finite queues coupled with a triggering scheduler. In the study of Markovian queueing systems, the emphasis had been on obtaining stationary system size probabilities; the transient behaviour has received considerably less attention as it is normally considered to be intractable. However, steady state measures do not reveal the complete picture of the system behaviour, because they ignore the transient start-up effects. Moreover, the steady state measures of system performance simply do not make sense because the system may never attain equilibrium (Whitt [17]). In many potential applications of queueing theory, the practitioner needs to know how the system will operate up to some time instant t. Further, if the system is empty initially, the fraction of time the server is busy the initial rate of output etc., will be below the steady state values hence the use of steady state results to obtain these measures is not appropriate. For instance, adaptive routing load balancing methods in networks require transient measures, such as queue length distribution, since information received from neighbouring nodes is always out-of-date. Thus, the investigation of the transient behaviour of the queueing system is also important from the point of view of theory as well as applications. Transient probabilities of a single batch service queueing system incorporating accessibility to the batches have been studied by Baburaj [2]. The busy period distribution of this system is studied expression for the mean busy period of the system is also obtained. The time-dependent solution of a single server Markovian queueing

3 Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes 65 system with service in batches of variable size has been investigated by Garg [7]. Further, probabilities of number of arrivals departures are also obtained by solving the difference equations recursively. In recent years, queueing systems with catastrophes have been investigated by Boucherie Boxma [3], Jain Sigman [9] Dudin Nishimura [6]. The notion of catastrophes occurring at rom, leading to annihilation of all the customers there the momentary inactivation of the service facilities until a new arrival of customers is not uncommon in many practical situations. The catastrophes may come either from outside the system or from another service station. In computer systems, if a job infected with virus arrives, it transmits virus to other processors inactivating files possibly the system itself. Hence, computer networks with a virus infection may be modelled as queueing networks with catastrophes. Comprehensive treatment of queueing models with catastrophes can be found in Gelenbe Pujolle [8], Chao et al. [4] Artalejo [1]. Although results have been reported seperately on queueing models with heterogenous servers queueing systems subject to catastrophes, no work has been found in the literature which studies queueing systems taking together the above mentioned features. Based on this observation, we have investigated the transient solution for the probabilities in the two-server queueing system subject to catastrophes, where one server is faster than the other, by defining a suitable probability generating function. The results of this paper are organized as follows: In section 2, we shall describe the model of two server heterogenous system with catastrophes obtain the timedependant state probabilities for the number in the system. Section 3 is devoted to steady state probabilities of the system size. In section 4, we present some important performance measures that are derived from the system size probabilities. 2. Model Description Analysis Consider an M/M/2 queueing system - server 1, the fast server server 2, the slow server. Assume that the service times follow exponential distributions with the service rate µ 1 for server 1 µ 2 for server 2 such that µ 1 > µ 2. Customer arrival process is Poisson with rate λ system has one waiting line. Each customer requires exactly one server for his service the queueing discipline is FCFS. When there are customers in the waiting line a server becomes free, the customer who is first in line

4 66 Information Management Sciences, Vol. 18, No. 1, March, 27 joins it. On the other h, a customer who arrives to an empty system joins the fast server with probability p the slow server with probability 1 p. Apart from arrival service processes, the catastrophes also occur at the service facilities in a Poisson manner with rate γ. Whenever a catastrophe occurs at the system, all the customers there are destroyed immediately, both the servers get inactivated momentarily the servers are ready for service when a new arrival occurs. Let {X(t), t R + } be the number of customers in the system at time t. Let P n (t) = P (X(t) = n), n = 2, 3, 4... denote the probability that there are n customers in the system at time t. Also let P, (t) = P (X(t) = ) be the probability that the system is empty at time t, P 1, (t) = P (X(t) = 1) be the probability that there is one customer in the system he is served by server 1 P,1 (t) = P (X(t) = 1) be the probability that there is one customer in the system he is served by server 2. From the above assumptions the state probabilities P, (t), P 1, (t), P,1 (t) P n (t), n = 2, 3, 4,... satisfy the following system of differential difference equations: dp, (t) = λp, (t) + µ 1 P 1, (t) + µ 2 P,1 (t) + γ(1 P, (t)) (2.1) dt dp 1, (t) = (λ + µ 1 + γ)p 1, (t) + λpp, (t) + µ 2 P 2 (t) (2.2) dt dp,1 (t) = (λ + µ 2 + γ)p,1 (t) + λ(1 p)p, (t) + µ 1 P 2 (t) (2.3) dt dp 2 (t) = (λ + µ 1 + µ 2 + γ)p 2 (t) + λp 1, (t) + λp,1 (t) + (µ 1 + µ 2 )P 3 (t) (2.4) dt dp n (t) = (λ + µ 1 + µ 2 + γ)p n (t) + λp n 1 (t) + (µ 1 + µ 2 )P n+1 (t), n = 3, 4, dt (2.5) We assume that there is no customer in the system at time t =, so that P, () = 1. We solve the above system of equations by using a probability generating function technique. Define the probability generating function P (z, t) = R (t) + P n+3 (t)z n+1 (2.6) n= where R (t) = P, (t) + P 1, (t) + P,1 (t) + P 2 (t), with the initial condition P (z, ) = 1.

5 Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes 67 Using the stard generating function argument, the system of equation (2.1) to (2.5) then yields P (z, t) t = where µ = µ 1 + µ 2. [λz + µ z (λ + µ + γ) ] [P (z, t) R (t)]+γ(1 R (t))+λ(z 1)P 2 (t) (2.7) Considering equation (2.7) as a first order linear differential equation in P (z, t) solving the same, we have, P (z, t) = e [λz+ µ z (λ+µ+γ)]t + { γ(1 R (u)) + λ(z 1)P 2 (u) (λz + µz ) } (λ + µ + γ) R (u) e [λz+ µ z (λ+µ+γ)](t u) du. (2.8) Using the Bessel function generating function (see Watson [16]), if α = 2 λµ β = λ µ, then e (λz+ µ z )t = n= I n (αt)(βz) n where I n (.) is the modified Bessel function of first kind of order n. Substituting this in equation (2.8), exping P (z, t) as a series in z comparing the co-efficient of z n on either side, we get, for n = 1, 2, 3,..., P n+2 (t) = e (λ+µ+γ)t β n I n (αt) + γβ n (1 R (u))i n (α(t u))e (λ+µ+γ)(t u) du for n = +β n 1 λ P 2 (u) [I n 1 (α(t u)) βi n (α(t u))] e (λ+µ+γ)(t u) du [ β n 1 R (u)e (λ+µ+γ)(t u) λi n 1 (α(t u)) + µβ 2 I n+1 (α(t u)) β(λ + µ + γ)i n (α(t u))] du (2.9) βr (t) = βe (λ+µ+γ)t I (αt) + βγ (1 R (u))i (α(t u))e (λ+µ+γ)(t u) du +λ P 2 (u)e (λ+µ+γ)(t u) [I 1 (α(t u)) βi (α(t u))] du R (u)e (λ+µ+γ)(t u) [2λI 1 (α(t u)) (λ + µ + γ)βi (α(t u))] du. (2.1)

6 68 Information Management Sciences, Vol. 18, No. 1, March, 27 As P (z, t) does not contain terms with negative powers of z, the right h side of (2.9) with n replaced by n must be zero. Thus, = β n e (λ+µ+γ)t I n (αt) + γβ n (1 R (u))e (λ+µ+γ)(t u) I n (α(t u)du +λβ n 1 P 2 (u)e (λ+µ+γ)(t u) [I n+1 (α(t u)) βi n (α(t u))] du [ β n 1 R (u)e (λ+µ+γ)(t u) λi n+1 (α(t u)) + µβ 2 I n 1 (α(t u)) where we have used I n (.) = I n (.). (λ + µ + γ)βi n (α(t u))] du (2.11) Using (2.11) in (2.9), after some algebra, we get, for n = 1, 2, 3,..., P n+2 (t) = nβ n P 2 (u)e (λ+µ+γ)(t u) I n(α(t u)) du. (2.12) t u Now, the probabilities P, (t), P 1, (t), P,1 (t) P 2 (t) remain to be found. For this, we consider the system of equations (2.1) -(2.3) subject to condition (2.1). Equations (2.1) - (2.3) can be expressed in matrix form as dp(t) dt = AP(t) + γe 1 + µ 2 P 2 (t)e 2 + µ 1 P 2 (t)e 3 (2.13) where (λ + γ) µ 1 µ 2 P(t) = (P, (t), P 1, (t), P,1 (t)) T, A = λp (λ + µ 1 + γ), λ(1 p) (λ + µ 2 + γ) e 1 = (1,, ) T, e 2 = (, 1, ) T e 3 = (,, 1) T. In the sequel, let P n (s) denote the Laplace transform of P n(t). Now, by taking Laplace transforms, the solution of (2.13) is obtained as with {( P (s) = (si A) γ ) } e 1 + µ 2 P2 s (s) e 2 + µ 1 P2 (s) e 3 (2.14) P() = (1,, ) T. (2.15) Thus, only P 2 (s) is to be found. We note that, if e = (1, 1, 1)T, R (s) = e T P (s) + P 2 (s). (2.16)

7 Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes 69 Taking Laplace transforms, after simplification, equation (2.1) yields ( R (s)(s + γ) = 1 + γ ) + 1 [ s 2 P 2 (s) w ] w 2 α 2 2λ (2.17) where w = s + λ + µ + γ. Let Using (2.17) in (2.16) solving for P2 (s), we get ( 1 + γ ) [ ] P2 s 1 (s + γ)e T (si A) 1 e 1 (s) = (s + γ)e T (si A) 1 [e 2 µ 2 + e 3 µ 1 ] + (s + λ + γ) 1 2 [w w 2 α 2 ]. (2.18) It is easy to see that (si A) 1 = where 1 D(s) (si A) 1 = ( ) a ij(s). 3 3 a 1 (s)a 2 (s) µ 1 a 2 (s) µ 2 a 1 (s) λpa 2 (s) b(s)a 2 (s) µ 2 λ(1 p) µ 2 λp (2.19) λ(1 p)a 1 (s) µ 1 λ(1 p) b(s)a 1 (s) λµ 1 p a 1 (s) = s + λ + µ 1 + γ, a 2 (s) = s + λ + µ 2 + γ, b(s) = s + λ + γ D(s) = s 3 + (3λ + 3γ + µ)s 2 + [(λ + µ 1 + γ)(λ + µ 2 + γ) +(λ + γ)(2(λ + γ) + µ)] s + (λ + γ)(λ + µ 1 + γ)(λ + µ 2 + γ). The characteristic roots of the matrix A are given by D(s) =. (2.2) By defining, a = 1 {3(λ + µ 1 + γ)(λ + µ 2 + γ) + 3(λ + γ)(2(λ + γ) + µ) (3(λ + γ) + µ) 2} 9 b = 1 { 2(3(λ + µ) + µ) 3 9(3(λ + γ) + µ) [(λ + µ 1 + γ)(λ + µ 2 + γ) 27 +(λ + γ)(2(λ + γ) + µ)] + 27(λ + γ)(λ + µ 1 + γ)(λ + µ 2 + γ)} n = 2 a θ = 1 3 cos 1 { b 2 a [ s i = n cos θ + (i 2) 2π 3 3 }, the characteristic roots of (2.2) are ] (3(λ + γ) + µ), i = 1, 2, 3. (2.21) 3

8 7 Information Management Sciences, Vol. 18, No. 1, March, 27 It is observed that a kj (s) are all rational algebraic functions of s. Hence, the inverse transform a kj (t) of a kj (s) can be obtained by partial fraction decompositions. Since the characteristic roots s i, i=1, 2, 3 of A are all real distinct, the inverse transform a kj (t) of a kj (s) are given below. a 11 (t) = a 12 (t) = a 13 (t) = a 21 (t) = a 22 (t) = a 23 (t) = a 31 (t) = a 32 (t) = a 33 (t) = Now using (2.19), we get (s m + λ + µ 1 + γ)(s m + λ + µ 2 + γ) Π 3 i=1,i m (s m s i ) µ 1 (s m + λ + µ 2 + γ) Π 3 i=1,i m (s m s i ) µ 2 (s m + λ + µ 1 + γ) Π 3 i=1,i m (s m s i ) λp(s m + λ + µ 2 + γ) Π 3 i=1,i m (s m s i ) e smt e smt e smt e smt [(s m + λ + µ 2 + γ)(s m + λ + γ) λµ 2 (1 p)] Π 3 i=1,i m (s m s i ) µ 2 λp 3i=1,i m (s m s i ) esmt λ(1 p)(s m + λ + µ 1 + γ) Π 3 i=1,i m (s m s i ) µ 1 λ(1 p) Π 3 i=1,i m (s m s i ) esmt e smt [(s m + λ + γ)(s m + λ + µ 1 + γ) λµ 1 (1 p)] Π 3 i=1,i m (s m s i ) e smt e smt. (s + γ)e T (si A) 1 e 1 = (s + γ) a j1(s) (2.22) j=1 (s + γ)e T (si A) 1 [µ 2 e 2 + µ 1 e 3 ] = (s + γ) µ 2 a j2(s) + µ 1 a j3(s). (2.23) j=1 j=1 Substituting (2.22) (2.23) in (2.18), we obtain P 2 (s) = ( 1 + γ ) s [1 (s + γ) 3j=1 a j1 [ (s)] (s+γ) µ 3j=1 2 a j2 (s)+µ 3j=1 1 a j3 ]+(s+λ+γ) (s) 1 2 [w w 2 α 2 ]. (2.24)

9 Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes 71 Using equation (2.19) in (2.14), we have P,(s) 1 { (s + γ) = (s+ vλ+µ 1 +γ)(s+λ+µ 2 +γ)+µ 1 µ 2 (s+λ+µ 2 +γ)p2 (s) D(s) s } +µ 1 µ 2 (s + λ + µ 1 + γ)p2 (s) (s + γ) = a 11 s (s) + [µ 2a 12 (s) + µ 1a 13 (s)]p 2 (s) (2.25) P1,(s) 1 + γ) = {λp(s (s + λ + µ 2 + γ) + µ 1 µ 2 λpp2 (s) + µ 2 [(s + λ + γ) D(s) s (s + λ + µ 2 + γ) λµ 2 (1 p)]p2 (s)} (s + γ) = a 21 s (s) + [µ 2a 22 (s) + µ 1a 23 (s)]p 2 (s) (2.26) P,1 (s) = 1 + γ) {λ(1 p)(s (s + λ + µ 1 + γ) + µ 1 µ 2 λ(1 p)p2 D(s) s (s) +µ 1 [(s + λ + γ)(s + λ + µ 1 + γ) λµ 1 p]p2 (s)} (s + γ) = a s 31(s) + [µ 2 a 32(s) + µ 1 a 33(s)]P2 (s). (2.27) By matrix theory, the characteristic roots s i, i=1, 2, 3 of A given in (2.21) are all real distinct. Defining s =, it can be shown by partial fraction decompositions that (s + γ) 2 a s 11(s) = 1 + (s + γ) 2 a 21 s (s) = (s + γ) 2 a s 31(s) = (s + γ)a 12(s) = m= m= m= (s + γ)a 22 (s) = 1 + (s m + λ + µ 1 + γ)(s m + λ + µ 2 + γ)(s m + γ) 2 Π 3 i=1,i m (s m s i )(s s m ) λp(s m + λ + µ 2 + γ)(s m + γ) 2 Π 3 i=1,i m (s m s i )(s s m ) = c 21 (s) λ(1 p)(s m + λ + µ 1 + γ)(s m + γ) 2 Π 3 i=1,i m (s m s i )(s s m ) (s m + γ)µ 1 (s m + λ + µ 2 + γ) Π 3 i=1,i m (s m s i )(s s m ) = 1 + c 22(s) (s + γ)a 32 (s) = (s + γ)a 13 (s) = (s + γ)a 33(s) = 1 + = c 12(s) = c 31(s) = 1 + c 11(s) (s m + γ)[(s m + λ + γ)(s m + λ + µ 2 + γ) λµ 2 (1 p)] Π 3 i=1,i m (s m s i )(s s m ) (s m + γ)λµ 1 (1 p) Π 3 i=1,i m (s m s i )(s s m ) = c 32 (s) (s m + γ)µ 2 (s m + λ + µ 1 + γ) Π 3 i=1,i m (s m s i )(s s m ) = c 23 (s) (s m + γ)[(s m + λ + γ)(s m + λ + µ 1 + γ) λµ 1 p] Π 3 i=1,i m (s m s i )(s s m ) = 1 + c 33(s)

10 72 Information Management Sciences, Vol. 18, No. 1, March, 27 where c ij (s) s, denote the summation terms in the above expressions. Using these in (2.22), after some algebraic manipulations, we get P 2 (s) = ( 2 α ) [ ] [ w w 2 α 2 γ s ] 3 j=1 c j1 (s) α [ ] w w 2 α 2 [µ 3j=1 α 2 2 c j2 (s) + µ 1 3j=1 c j3 (s) ] so that ( ) [ 2 P2 (s) = w ] [γ w 2 α 2 ] [ ( ) ] 1 w w α α s h 1 (s) α 2 α 2 [µ 2 h 2 (s)+µ 1h 3 (s)] (2.28) where h i (s) = j=1 c ji (s), i = 1, 2, 3. The above can be expressed as P2 (s) = ( 1) n 2n+1 γ ( ) w w 2 α 2 n+1 ( ) w w α n= n+1 h s α 1(s) 2 α 2 n+1 α [µ 2 h 2(s) + µ 1 h 3(s)] n which yields ( ) 2 n+1 n P2 (s) = ( 1) n n µ k α 2µ 1 n k (h 2(s)) k (h 3(s)) n k n= k= k [w ] n+1 w 2 α [w 2 ] n+1 w 2 α 2 γ s α n+1 h 1(s) α n+1. On inversion, we get an explicit expression for P 2 (t) as P 2 (t) = γ n= ( 2 n+1 ( 1) α) n (n+1) n n k= k µ k 2µ n k 1 u v h (k) 2 (y)h(n k) 3 (v y)dy I n+1(α(u v)) e (λ+µ+γ)(u v) dvdu (u v) ( ) 2 n+1 n + ( 1) n+1 (n + 1) n u µ k α 2µ 1 n k h (k) 2 (u v) k n= v h (n k) k= 3 (v y)h 1 (y)dydv I n+1(α(t u)) e (λ+µ+γ)(t u) du (2.29) (t u)

11 Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes 73 where h (n) i (t) is the n-fold convolution of h i (t) with itself. We note that h i () = δ(t), the Dirac delta function. Inverting (2.25) - (2.27), after some algebraic manipulation, we get, P, (t) = a 11 (t) + γ P 1, (t) = a 21 (t) + γ P,1 (t) = a 31 (t) + γ a 11 (u)du + a 21 (u)du + a 31 (u)du + [µ 2 a 12 (t u) + µ 1 a 13 (t u)] P 2 (u)du (2.3) [µ 2 a 22 (t u) + µ 1 a 23 (t u)] P 2 (u)du (2.31) [µ 2 a 32 (t u) + µ 1 a 33 (t u)] P 2 (u)du. (2.32) Thus (2.12) (2.29) -(2.32) completely determine all the system size probabilities. 3. Steady State Probabilities In this section, we shall discuss the structure of the steady state probabilities of the M/M/2 queueing system with heterogeneous servers disasters. Theorem 3.1. The steady state distribution of the M/M/2 queue with heterogeneous service rates catastrophes is obtained as follows: (i) For γ > λ µ, then γλ 2 [(µ 2 µ 1 )p + (λ + µ 1 + γ)] P 2 = γ [ (3.1) (λ + µ 1 + γ)(λ + µ 2 + γ)µ λµ 2 2 (1 p) + λµ 1µ 2 µ 2 1 λp] [(λ + γ µ) + ] (λ + µ + γ) 2 α 2 [(λ + γ)(λ + µ 1 + γ) (λ + µ 2 + γ) λµ 1 p(λ + µ 2 + γ) λµ 2 (1 p)(λ + µ 1 + γ)] ( ) β n [ ] n P n+2 = (λ + µ + γ) (λ + µ + γ) α 2 α 2 P 2, n = 1, 2, 3,... (3.2) P, = γ(λ + µ 1 + γ)(λ + µ 2 + γ) + µ 1 µ 2 (2(λ + γ) + µ)p 2 (λ+γ)(λ+µ 1 +γ)(λ+µ 2 +γ) λµ 1 p(λ+µ 2 +γ) λµ 2 (1 p)(λ+µ 1 +γ) (3.3) P 1, = γλp(λ+µ 2+γ)+{µ 2 [(λ+γ)(λ+µ 2 +γ) λµ 2 (1 p)]+µ 1 µ 2 λp} P 2 (3.4) (λ+γ)(λ+µ 1 +γ)(λ+µ 2 +γ) λµ 1 p(λ+µ 2 +γ) λµ 2 (1 p)(λ+µ 1 +γ) P,1 = γλ(1 p)(λ+µ 1+γ)+{µ 1 µ 2 λ(1 p)+µ 1 [(λ+γ)(λ+µ 1 +γ) λpµ 1 ]} P 2 (λ+γ)(λ+µ 1 +γ)(λ+µ 2 +γ) λµ 1 p(λ+µ 2 +γ) λµ 2 (1 p)(λ+µ 1 +γ). (3.5) (ii) For γ > λ = µ, then γµ 2 [(µ 2 µ 1 )p + (µ + µ 1 + γ)] P 2 = γ [ (µ + µ 1 + γ)(µ + µ 2 + γ)µ µµ 2 2 (1 p) + µµ 1µ 2 µ 2 1 µp] [γ + ] (2µ + γ) 2 4µ 2 [(µ + γ)(µ + µ 1 + γ)(µ + µ 2 + γ) µµ 1 p(µ + µ 2 + γ) µµ 2 (1 p)(µ + µ 1 + γ)] (3.6)

12 74 Information Management Sciences, Vol. 18, No. 1, March, 27 ( ) 1 n [ ] n P n+2 = (2µ + γ) (2µ + γ) 2µ 4µ 2 P 2, n = 1, 2, 3,... (3.7) γ(µ + µ 1 + γ)(µ + µ 2 + γ) + µ 1 µ 2 (3µ + 2γ))P 2 P, = (µ+γ)(µ+µ 1 +γ)(µ+µ 2 +γ) µµ 1 p(µ+µ 2 +γ) µµ 2 (1 p)(µ+µ 1 +γ) (3.8) P 1, = γµp(µ+µ 2+γ)+{µ 2 [(µ+γ)(µ+µ 2 +γ) µµ 2 (1 p)]+µ 1 µ 2 µp} P 2 (µ+γ)(µ+µ 1 +γ)(µ+µ 2 +γ) µµ 1 p(µ+µ 2 +γ) µµ 2 (1 p)(µ+µ 1 +γ) (3.9) P,1 = γµ(1 p)(µ+µ 1+γ) + {µ 1 µ 2 µ(1 p)+µ 1 [(µ+γ)(µ+µ 1 +γ) µpµ 1 ]} P 2 (µ+γ)(µ+µ 1 +γ)(µ+µ 2 +γ) µµ 1 p(µ+µ 2 +γ) µµ 2 (1 p)(µ+µ 1 +γ). (3.1) Proof. For γ > λ µ, from (2.24), we have (s + γ) [1 (s + γ) ] 3 P2 j=1 a j1 (s) = (s) { [ s (s + γ) µ 3j=1 2 a j2 (s)+µ ] 3j=1 1 a j3 (s) +(s+λ+γ) 1 2 [w ]}. w 2 α 2 Multiplying the above equation by s on both sides taking limit as s, we get lim sp γλ 2 [(µ 2 µ 1 )p + (λ + µ 1 + γ)] 2 (s) = s γ [. (3.11) (λ + µ 1 + γ)(λ + µ 2 + γ)µ λµ 2 2 (1 p) + λµ 1µ 2 µ 2 1 λp] [(λ + γ µ) + ] (λ + µ + γ) 2 α 2 [(λ + γ)(λ + µ 1 + γ) (λ + µ 2 + γ) λµ 1 p(λ + µ 2 + γ) λµ 2 (1 p)(λ + µ 1 + γ)] The result (3.1) follows directly from (3.11), by using Tauberian theorem. Taking Laplace transform of (2.12), we have ( ) β n [ ] n Pn+2 (s) = (s+λ+µ+γ) (s+λ+µ+γ) α 2 α 2 P2 (s), n = 1, 2, 3,.... (3.12) As before, multiplying (3.12) by s on both sides taking limit as s, we get ( ) β n [ ] n lim sp n+2 (s) = lim (s + λ + µ + γ) (s + λ + µ + γ) s s α 2 α 2 sp2 (s), n = 1, 2, 3,... (3.13) which yields (3.2), by applying Tauberian theorem again. Similarly, the results (3.3) - (3.5) can be obtained from (2.25) - (2.27) respectively. For γ > λ = µ, the results (3.6) - (3.1) can be obtained directly by putting λ = µ in (3.1) - (3.5). Remark 1. It is observed that the steady state probabilities of this queueing model exist if only if γ > or γ = λ < µ 1 + µ 2. Rubinovitch [14] has obtained the steady state probabilities of the M/M/2 heterogeneous queueing model without disasters under the steady state condition λ < µ 1 + µ 2. It is interesting to note that, for γ = λ < µ 1 + µ 2, our results (3.1) - (3.5) agree with Rubinovitch s results.

13 Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes 75 Remark 2. It is observed that if p = 1, the customer arriving at an empty system always joins the fast server if p = 1/2, the customer arriving at an empty system joins one of the servers with equal probability. 4. Performance Measures A number of interesting performance measures are studied in this section, including the average number of customers in the system, the probability that an arriving customer is required to join the queue, the probability that the system has n (n = 1, 2) busy servers, the mean number of busy servers, the mean busy period of the system so forth. The mean number of customers in the system: Let X(t) be the number of customers in the system at time t. The average number of customers in the system at time t is given by E(X(t)) = P 1, (t) + P,1 (t) + (n + 2)P n+2 (t). n= Using (2.12), (2.31) (2.32), the above can be written as E(X(t)) = a 21 (t) + γ +a 31 (t) + γ +2P 2 (t) + n=1 where P 2 (t) is given in (2.29). as a 21 (u)du + a 31 (u)du + (n + 1)nβ n [µ 2 a 22 (t u) + µ 1 a 23 (t u)]p 2 (u)du [µ 2 a 32 (t u) + µ 1 a 33 (t u)]p 2 (u)du P 2 (u)e (λ+µ+γ)(t u) I n(α(t u)) du (4.1) (t u) If γ >, the mean number of customers in the system under steady state is computed E(X) = 2µ{4µ [(λ + µ + γ) (λ + µ + γ) 2 4λµ]}P 2 {2µ [(λ + µ + γ) (λ + µ + γ) 2 4λµ]} 2 γ[λp(µ 2 µ 1 ) + λ(λ + µ 1 + γ)] + P 2 {µ 2 (γ + λ)[(λ + µ 2 + γ) +µ 1 λ λ(1 p)] + µ 1 [(λ + γ)(λ + µ 1 + γ) µ 1 λp]} +, (λ + γ)(λ + µ 1 + γ)(λ + µ 2 + γ) λµ 1 p(λ + µ 2 + γ) λµ 2 (1 p)(λ + µ 1 + γ) if λ µ (4.2)

14 76 Information Management Sciences, Vol. 18, No. 1, March, 27 E(X) = 2µ{2µ γ + 4µγ + γ 2 }P 2 { 4µγ + γ 2 γ} 2 γ[µp(µ 2 µ 1 ) + µ(µ + µ 1 + γ)] + P 2 {µ 2 (γ + µ)[(µ + µ 2 + γ) +µ 1 µ µ(1 p)] + µ 1 [(µ + γ)(µ + µ 1 + γ) µ 1 µp]} +, (µ + γ)(µ + µ 1 + γ)(µ + µ 2 + γ) µµ 1 p(µ + µ 2 + γ) µµ 2 (1 p)(µ + µ 1 + γ) where P 2 is given in (3.1) for λ µ in (3.6) for λ = µ. Probability of arriving customers joining the queue: if λ = µ (4.3) The probability that an arriving customer is required to join the queue at time t is given by P (X(t) 2) = P n+2 (t) n= = P 2 (t) + nβ n n=1 P 2 (u)e (λ+µ+γ)(t u) I n(α(t u)) du. (4.4) (t u) Similarly, for γ >, the steady state probability that an arriving customer joins the queue is 2µP 2, if λ µ (µ λ γ)+ (λ+µ+γ) P (X 2) = P n+2 = 2 4λµ n= 2µP 2, if λ = µ. γ 2 +4γµ γ (4.5) The number of busy servers: Let M(t) denote the number of busy servers at time t. system has n busy servers is given as, P (X(t) = 1) = P 1, (t) + P,1 (t), for n = 1 P {M(t) = n} = P (X(t) > 1) = n= P n+2 (t), for n = 2 The probability that the (4.6) the corresponding steady state probability is obtained for γ > λ µ as γ[λp(µ 2 µ 1 )+λ(µ 1 +λ+γ)]+p 2 {µ 2 [(λ+γ)(λ+µ 2 +γ) λ(1 p)+µ 1 λ]+µ 1 [(λ+γ)(λ+µ 1 +γ) µ 1 λp]}, if n = 1 P (M = n) = [(λ + γ)(µ 1 + λ + γ)(λ + µ 2 + γ) λµ 1 p(λ + µ 2 + γ) (4.7) λµ 2 (1 p)(λ + µ 1 + γ)] 2µP 2 {µ λ γ+ (λ+µ+γ) 2 4λµ}, if n = 2.

15 Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes 77 Similarly the above probability can be obtained directly, for γ > λ = µ, by substituting λ = µ in (4.7). Furthermore, the mean number of busy servers at time t is given by E(M(t)) = P 1, (t) + P,1 (t) + 2 P n+2 (t) which can be simplified as E(M(t)) = 2[1 P, (t)] [P 1, (t) + P,1 (t)]. (4.8) For γ >, the corresponding steady state result is given as 2[λ(λ + µ 1 + γ)(λ + µ 2 + γ) λµ 1 p(λ + µ 2 + γ) λµ 2 (1 p)(λ + µ 1 + γ) µ 1 µ 2 (2(λ + γ) + µ)]p 2 γ[λp(µ 2 µ 1 ) + λ(λ + µ 1 + γ)]]+{[µ 2 ((λ + γ) (λ + µ 2 + γ)) λ(1 p) + µ 1 λ] + µ 1 [(λ + γ)(λ + µ 1 + γ) µ 1 λp]}p 2 E(M) =, (λ + γ)(λ + µ 1 + γ)(λ + µ 2 + γ) λµ 1 p(λ + µ 2 + γ) λ µ 2 (1 p)(λ + µ 1 + γ) if λ µ (4.9) 2[µ(µ + µ 1 + γ)(µ + µ 2 + γ) µµ 1 p(µ + µ 2 + γ) µµ 2 (1 p)(µ + µ 1 + γ) µ 1 µ 2 ((3µ + 2γ))]P 2 γ[µp(µ 2 µ 1 ) + µ(µ + µ 1 + γ)]] + {[µ 2 ((µ + γ) (µ + µ 2 + γ)) µ(1 p) + µ 1 µ] + µ 1 [(µ + γ)(µ + µ 1 + γ) µ 1 µp]}p 2 E(M) =, (µ + γ)(µ + µ 1 + γ)(µ + µ 2 + γ) µµ 1 p(µ + µ 2 + γ) µ µ 2 (1 p)(µ + µ 1 + γ) if λ = µ. (4.1) The mean of the system busy period: Another interesting performance measure in queueing theory context is the mean of the system busy period. The system busy period L is defined as the period that starts at an epoch when an arriving customer finds an empty system ends at the next departure epoch at which the system is empty. The mean length of the system busy period of our model is obtained in a direct way by the theory of regenerative processes which leads to the limiting probability. P, = lim P (X(t) = ) = E(T ) t 1 λ + E(L) n=

16 78 Information Management Sciences, Vol. 18, No. 1, March, 27 where T is the amount of time in a regenerative cycle during which the system is in the state zero (the system size is zero). It is clear that so that which yields E(T ) = 1 λ E(L) = 1 1 (P, λ 1), E(L) = λ(λ + γ) {(1 p)(µ 1 µ 2 ) + (λ + µ 2 + γ)} µ 1 µ 2 (2(λ + γ) + µ)p 2 λγ(λ + µ 1 + γ)(λ + µ 2 + γ) + λµ 1 µ 2 (2(λ + γ) + µ)p 2, if λ µ (4.11) E(L) = µ(µ + γ) {(1 p)(µ 1 µ 2 ) + (µ + µ 2 + γ)} µ 1 µ 2 (2γ + 3µ)P 2 µγ(µ + µ 1 + γ)(µ + µ 2 + γ) + µµ 1 µ 2 (3µ + 2γ)P 2, if λ = µ. (4.12) Finally, the mean number of customers E(N), served during the busy period is computed as E(N) = 1 + λe(l) so that (λ + γ)(λ + µ 1 + γ)(λ + µ 2 + γ) λµ 1 p(λ + µ 2 + γ) λµ 2 (1 p)(λ + µ 1 + γ) E(N) =, if λ µ (4.13) γ(λ + µ 1 + γ)(λ + µ 2 + γ) + µ 1 µ 2 (2(λ + γ) + µ)p 2 (µ + γ)(µ + µ 1 + γ)(µ + µ 2 + γ) µµ 1 p(µ + µ 2 + γ) µµ 2 (1 p)(µ + µ 1 + γ) E(N) =, if λ = µ. (4.14) γ(µ + µ 1 + γ)(µ + µ 2 + γ) + µ 1 µ 2 (2γ + 3µ)P 2 5. Conclusion In the foregoing analysis, a two server heterogeneous queueing system with catastrophes is considered to obtain the time-dependent probabilities for the number of customers

17 Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes 79 in the system. The steady state probabilities of the system size are also studied. Finally, some important performance measures have been obtained from the steady state probabilities. Acknowledgements The authors thank the anonymous referees for their valuable comments suggestions to improve the presentation of the paper. References [1] J. R. Artalejo, G-Networks: A versatile approach for work removal in queueing networks, European Journal of Operations Research, Vol.126, No.2, , 2. [2] C. Baburaj, On the transient distribution of a single batch service queueing system with accessibility to the batches, International Journal of Information Management Sciences, Vol. 11, No. 2, 27-36, 2. [3] R. J. Boucherie O. J. Boxma, The workload in the M/G/1 queue with work removal, Probability in Engineering Informational Sciences, Vol. 1, , [4] X. Chao, M. Miyazawa M. Pinedo, Queueing Networks, Customers, Signals Product form Solutions, John Wiley Sons, Chichester, [5] V. Dörrsam, Materialfluβorientierte Leistungsanalyse einstufiger Produktionssysteme, Dissertation, Universität Karlsruhe (in German), [6] A. N. Dudin S. Nishimura, A BMAP/SM/1 queueing system with Markovian arrival input of disasters, Journal of Applied Probability, Vol.36, , [7] P. C. Garg, A measure for time dependent queueing problem with service in batches of variable size, International Journal of Information Management Sciences, Vol. 14, No. 4, 83-87, 23. [8] E. Gelenbe G. Pujolle, Introduction to Queueing Networks, (2nd edition), John Wiley Sons, Chichester, [9] G. Jain K. Sigman, A Pollaczek-Khinchine formula for M/G/1 queues with disasters, Journal of Applied Probability, Vol. 33, , [1] S. S. Lavenberg, A perspective on queueing models of computer performance in queueing theory its applications, Liber Amicorum for J. N. Cohen; CWI Monograph 7, [11] E. D. Lazowaska, G. S. Zahorjan K. C. Sevcik, Quantitative system performance, Prentice-Hall, Englewood Cliffs, [12] W. Liu P. Kumar, Optimal control of a queueing system with two heterogeneous servers, IEEE Transactions on Automatic Contol, Vol. 29, , [13] M. Mittler, The variability of cycle times in semiconductor manufacturing, Dissertation, Universität Würzburg, [14] M. Rubinovitch, The slow server problem, Journal of Applied Probability, Vol. 22, , [15] V. P. Singh, Two-server Markovian queues with balking: Heterogeneous vs. homogeneous servers, Operations Research, Vol.19, , 197. [16] G. N. Watson, A Treaties on the Theory of Bessel Functions, Cambridge University Press, Cambridge, [17] W. Whitt, Untold horrors of the waiting room: What the equilibrium distribution will never tell about th queue length process, Management Science, Vol. 29, , 1983.

18 8 Information Management Sciences, Vol. 18, No. 1, March, 27 Authors Information B. Krishna Kumar is an Assistant Professor in the Department of Mathematics, Anna University, India. He received his M.Sc. M.Phil degrees from University of Madras in respectively. Subsequently, he has obtained his Ph.D degree from I. I. T., Madras in He had been in N.T.T., Multimedia Laboratory, Tokyo as a post-doctoral fellow during His research interests include Queueing Models Their Applications, Branching Processes Mathematical Ecology. Dr. B. Krishna Kumar is a member of the Indian Society for Probability Statistics the Operations Research Society of India. Department of Mathematics, Anna University, Chennai, India. drbkkumar@hotmail.com TEL : S. Pavai Madheswari is an Assistant Professor in the Department of Mathematics, R. M. K Engineering College affiliated to Anna University, India. She received her Ph.D in Mathematics from Anna University. Her research interests include Queueing Theory Stochastic Modelling of Communication systems. Department of Mathematics, R. M. K. Engineering College, Kavaraipettai, Thiruvallur Dt., India. pavai arun@yahoo.com TEL : K. S. Venkatakrishnan is a Professor in the Department of Mathematics, Anna University, India. He is with Anna University for the past 26 years. His research interests include Reliability Analysis, Operations Research, Queueing Theory Stochastic Dynamic Systems. He is a member of the Operations Research Society of India. Department of Mathematics, Anna University, Chennai, India.