Information and Management Sciences Volume 18, Number 1, pp. 63-80, 2007
Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes B. Krishna Kumar
S. Pavai Madheswari
Anna University
Anna University
India
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 and computer networks. Computer systems or data transmission networks deal with systems having multiple resources ( Central processing units, Channels, Memories etc.). Any 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 [10]. 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 2005; Revised February 2006; Accepted June 2006. Supported by ours.
64
Information and Management Sciences, Vol. 18, No. 1, March, 2007
main feature of the operation of almost any manufacturing system. The role of quality and service performance are crucial aspects in customer perceptions and firms must dedicate special attention to them when designing and 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 and specially in queueing theory. A Markovian queueing system with balking and 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 and 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] and D¨orrsam [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 and 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 and the initial rate of output etc., will be below the steady state values and hence the use of steady state results to obtain these measures is not appropriate. For instance, adaptive routing and 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 and batch service queueing system incorporating accessibility to the batches have been studied by Baburaj [2]. The busy period distribution of this system is studied and expression for the mean busy period of the system is also obtained. The time-dependent solution of a single server Markovian queueing
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 and departures are also obtained by solving the difference equations recursively. In recent years, queueing systems with catastrophes have been investigated by Boucherie and Boxma [3], Jain and Sigman [9] and Dudin and Nishimura [6]. The notion of catastrophes occurring at random, leading to annihilation of all the customers there and 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 and 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 and Pujolle [8], Chao et al. [4] and Artalejo [1]. Although results have been reported seperately on queueing models with heterogenous servers and 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 and 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 and Analysis Consider an M/M/2 queueing system - server 1, the fast server and server 2, the slow server. Assume that the service times follow exponential distributions with the service rate µ1 for server 1 and µ2 for server 2 such that µ1 > µ2 . Customer arrival process is Poisson with rate λ and system has one waiting line. Each customer requires exactly one server for his service and the queueing discipline is FCFS. When there are customers in the waiting line and a server becomes free, the customer who is first in line
66
Information and Management Sciences, Vol. 18, No. 1, March, 2007
joins it. On the other hand, a customer who arrives to an empty system joins the fast server with probability p and the slow server with probability 1 − p. Apart from arrival and 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 and 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 P0,0 (t) = P (X(t) = 0) be the probability that the system is empty at time t, P 1,0 (t) = P (X(t) = 1) be the probability that there is one customer in the system and he is served by server 1 and P0,1 (t) = P (X(t) = 1) be the probability that there is one customer in the system and he is served by server 2. From the above assumptions the state probabilities P 0,0 (t), P1,0 (t), P0,1 (t) and Pn (t), n = 2, 3, 4, . . . satisfy the following system of differential difference equations: dP0,0 (t) dt dP1,0 (t) dt dP0,1 (t) dt dP2 (t) dt and dPn (t) dt
= −λP0,0 (t) + µ1 P1,0 (t) + µ2 P0,1 (t) + γ(1 − P0,0 (t))
(2.1)
= −(λ + µ1 + γ)P1,0 (t) + λpP0,0 (t) + µ2 P2 (t)
(2.2)
= −(λ + µ2 + γ)P0,1 (t) + λ(1 − p)P0,0 (t) + µ1 P2 (t)
(2.3)
= −(λ + µ1 + µ2 + γ)P2 (t) + λP1,0 (t) + λP0,1 (t) + (µ1 + µ2 )P3 (t)
(2.4)
= −(λ + µ1 + µ2 + γ)Pn (t) + λPn−1 (t) + (µ1 + µ2 )Pn+1 (t), n = 3, 4, 5... . (2.5)
We assume that there is no customer in the system at time t = 0, so that P 0,0 (0) = 1. We solve the above system of equations by using a probability generating function technique. Define the probability generating function P (z, t) = R0 (t) +
∞ X
Pn+3 (t)z n+1
n=0
where R0 (t) = P0,0 (t) + P1,0 (t) + P0,1 (t) + P2 (t), with the initial condition P (z, 0) = 1.
(2.6)
Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes
67
Using the standard generating function argument, the system of equation (2.1) to (2.5) then yields ∂P (z, t) µ = λz + − (λ + µ + γ) [P (z, t) − R0 (t)] + γ(1 − R0 (t)) + λ(z − 1)P2 (t) (2.7) ∂t z
where µ = µ1 + µ2 . Considering equation (2.7) as a first order linear differential equation in P (z, t) and solving the same, we have, µ
P (z, t) = e[λz+ z −(λ+µ+γ)]t Z t
+
0
µ γ(1 − R0 (u)) + λ(z − 1)P2 (u) − λz + − (λ + µ + γ) R0 (u) z µ
× e[λz+ z −(λ+µ+γ)](t−u) du.
(2.8)
√ Using the Bessel function generating function (see Watson [16]), if α = 2 λµ and β = q
λ µ,
then e(λz+ z )t = µ
∞ X
In (αt)(βz)n
n=−∞
where In (.) is the modified Bessel function of first kind of order n. Substituting this in equation (2.8), expanding P (z, t) as a series in z and comparing the co-efficient of z n on either side, we get, for n = 1, 2, 3, . . . , Pn+2 (t) = e
−(λ+µ+γ)t n
β In (αt) + γβ
+β n−1 λ −β n−1
Z
Z
t
0 t
0
n
Z
t 0
(1 − R0 (u))In (α(t − u))e−(λ+µ+γ)(t−u) du
P2 (u) [In−1 (α(t − u)) − βIn (α(t − u))] e−(λ+µ+γ)(t−u) du h
R0 (u)e−(λ+µ+γ)(t−u) λIn−1 (α(t − u)) + µβ 2 In+1 (α(t − u)) −β(λ + µ + γ)In (α(t − u))] du
(2.9)
and for n = 0 βR0 (t) = βe
−(λ+µ+γ)t
+λ −
Z
Z
t
0 t
0
I0 (αt) + βγ
Z
t 0
(1 − R0 (u))I0 (α(t − u))e−(λ+µ+γ)(t−u) du
P2 (u)e−(λ+µ+γ)(t−u) [I1 (α(t − u)) − βI0 (α(t − u))] du
R0 (u)e−(λ+µ+γ)(t−u) [2λI1 (α(t − u)) − (λ + µ + γ)βI0 (α(t − u))] du. (2.10)
68
Information and Management Sciences, Vol. 18, No. 1, March, 2007
As P (z, t) does not contain terms with negative powers of z, the right hand side of (2.9) with n replaced by −n must be zero. Thus, n −(λ+µ+γ)t
0=β e
+λβ n−1 −β
n−1
Z
Z
t
0 t
0
In (αt) + γβ
n
Z
t 0
(1 − R0 (u))e−(λ+µ+γ)(t−u) In (α(t − u)du
P2 (u)e−(λ+µ+γ)(t−u) [In+1 (α(t − u)) − βIn (α(t − u))] du h
R0 (u)e−(λ+µ+γ)(t−u) λIn+1 (α(t − u)) + µβ 2 In−1 (α(t − u)) −(λ + µ + γ)βIn (α(t − u))] du
(2.11)
where we have used I−n (.) = In (.). Using (2.11) in (2.9), after some algebra, we get, for n = 1, 2, 3, . . ., Pn+2 (t) = nβ n
Z
t 0
P2 (u)e−(λ+µ+γ)(t−u)
In (α(t − u)) du. t−u
(2.12)
Now, the probabilities P0,0 (t), P1,0 (t), P0,1 (t) and P2 (t) remain to be found. For this, we consider the system of equations (2.1) -(2.3) subject to condition (2.10). Equations (2.1) - (2.3) can be expressed in matrix form as dP(t) = AP(t) + γe1 + µ2 P2 (t)e2 + µ1 P2 (t)e3 dt where
−(λ + γ)
P(t) = (P0,0 (t), P1,0 (t), P0,1 (t))T , A =
λp
λ(1 − p)
(2.13)
µ1
µ2
−(λ + µ1 + γ)
0
0
−(λ + µ2 + γ)
e1 = (1, 0, 0)T , e2 = (0, 1, 0)T and e3 = (0, 0, 1)T .
,
In the sequel, let Pn∗ (s) denote the Laplace transform of P n (t). Now, by taking Laplace transforms, the solution of (2.13) is obtained as P ∗ (s) = (sI − A)−1
1+
γ e1 + µ2 P2∗ (s) e2 + µ1 P2∗ (s) e3 s
(2.14)
with P(0) = (1, 0, 0)T .
(2.15)
Thus, only P2∗ (s) is to be found. We note that, if e = (1, 1, 1) T , R0∗ (s) = eT P ∗ (s) + P2∗ (s).
(2.16)
Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes
69
Taking Laplace transforms, after simplification, equation (2.10) yields
R0∗ (s)(s + γ) = 1 +
γ s
where w = s + λ + µ + γ.
h i p 1 + P2∗ (s) w − w2 − α2 − 2λ 2
(2.17)
Using (2.17) in (2.16) and solving for P 2∗ (s), we get P2∗ (s) =
1+
γ s
h
1 − (s + γ)eT (sI − A)−1 e1
i
(s + γ)eT (sI − A)−1 [e2 µ2 + e3 µ1 ] + (s + λ + γ) − 21 [w −
√ . (2.18) w 2 − α2 ]
Let
(sI − A)−1 = a∗ij (s) It is easy to see that
(sI − A)−1 = where
3×3
.
a (s)a2 (s) µ1 a2 (s) 1 λpa2 (s) b(s)a2 (s) − µ2 λ(1 − p)
1 | D(s) |
λ(1 − p)a1 (s)
a1 (s) = s + λ + µ1 + γ,
µ1 λ(1 − p)
a2 (s) = s + λ + µ2 + γ,
µ2 a1 (s) µ2 λp b(s)a1 (s) − λµ1 p
(2.19)
b(s) = s + λ + γ
and | D(s) | = s3 + (3λ + 3γ + µ)s2 + [(λ + µ1 + γ)(λ + µ2 + γ) +(λ + γ)(2(λ + γ) + µ)] s + (λ + γ)(λ + µ1 + γ)(λ + µ2 + γ). The characteristic roots of the matrix A are given by | D(s) |= 0.
(2.20)
By defining, o 1n 3(λ + µ1 + γ)(λ + µ2 + γ) + 3(λ + γ)(2(λ + γ) + µ) − (3(λ + γ) + µ) 2 9 1 n b= 2(3(λ + µ) + µ)3 − 9(3(λ + γ) + µ) [(λ + µ1 + γ)(λ + µ2 + γ) 27 +(λ + γ)(2(λ + γ) + µ)] + 27(λ + γ)(λ + µ1 + γ)(λ + µ2 + γ)}
a=
√ n = 2 −a and θ =
1 3
b cos−1 {− 2√−a }, the characteristic roots of (2.20) are 3
si = n cos θ + (i − 2)
2π (3(λ + γ) + µ) , i = 1, 2, 3. − 3 3
(2.21)
70
Information and Management Sciences, Vol. 18, No. 1, March, 2007
It is observed that a∗kj (s) are all rational algebraic functions of s. Hence, the inverse transform akj (t) of a∗kj (s) can be obtained by partial fraction decompositions. Since the characteristic roots si , i=1, 2, 3 of A are all real and distinct, the inverse transform a kj (t) of a∗kj (s) are given below. a11 (t) = a12 (t) = a13 (t) = a21 (t) = a22 (t) = a23 (t) = a31 (t) =
3 X (sm + λ + µ1 + γ)(sm + λ + µ2 + γ)
Π3i=1,i6=m (sm − si )
m=1 3 X
e sm t
µ1 (sm + λ + µ2 + γ) sm t e Π3i=1,i6=m (sm − si ) m=1 3 X µ2 (sm + λ + µ1 + γ)
m=1 3 X
Π3i=1,i6=m (sm − si )
e sm t
λp(sm + λ + µ2 + γ) sm t e Π3i=1,i6=m (sm − si ) m=1
3 X [(sm + λ + µ2 + γ)(sm + λ + γ) − λµ2 (1 − p)]
m=1 3 X
m=1 3 X
Π3i=1,i6=m (sm − si )
µ2 λp e sm t (s − s ) i i=1,i6=m m
Q3
λ(1 − p)(sm + λ + µ1 + γ) sm t e Π3i=1,i6=m (sm − si ) m=1 3 X
a32 (t) =
µ1 λ(1 − p) 3 Π (s − si ) m=1 i=1,i6=m m
a33 (t) =
3 X [(sm + λ + γ)(sm + λ + µ1 + γ) − λµ1 (1 − p)]
and
e sm t
e sm t
Π3i=1,i6=m (sm − si )
m=1
e sm t .
Now using (2.19), we get (s + γ)eT (sI − A)−1 e1 = (s + γ)
3 X
a∗j1 (s)
(2.22)
j=1
and
(s + γ)eT (sI − A)−1 [µ2 e2 + µ1 e3 ] = (s + γ) µ2
3 X
a∗j2 (s) + µ1
j=1
Substituting (2.22) and (2.23) in (2.18), we obtain P2∗ (s)
=
1+ h
(s+γ) µ2
P3
γ s [1
∗ j=1 aj2 (s)+µ1
− (s + γ)
P3
∗ j=1 aj1 (s)]
3 X
j=1
a∗j3 (s) . (2.23)
. (2.24) √ ∗ (s) +(s+λ+γ)− 1 [w− w 2 −α2 ] a j=1 j3 2
P3
i
Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes
71
Using equation (2.19) in (2.14), we have ∗ P0,0 (s) =
1 n (s + γ) (s+ vλ+µ1 +γ)(s+λ+µ2 +γ)+µ1 µ2 (s+λ+µ2 +γ)P2∗ (s) | D(s) | s
+µ1 µ2 (s + λ + µ1 + γ)P2∗ (s)
o
(s + γ) ∗ a11 (s) + [µ2 a∗12 (s) + µ1 a∗13 (s)]P2∗ (s) (2.25) s 1 (s + γ) ∗ P1,0 (s) = {λp (s + λ + µ2 + γ) + µ1 µ2 λpP2∗ (s) + µ2 [(s + λ + γ) | D(s) | s ×(s + λ + µ2 + γ) − λµ2 (1 − p)]P2∗ (s)} (s + γ) ∗ a21 (s) + [µ2 a∗22 (s) + µ1 a∗23 (s)]P2∗ (s) (2.26) = s and (s + γ) 1 ∗ {λ(1 − p) (s + λ + µ1 + γ) + µ1 µ2 λ(1 − p)P2∗ (s) P0,1 (s) = | D(s) | s +µ1 [(s + λ + γ)(s + λ + µ1 + γ) − λµ1 p]P2∗ (s)} (s + γ) ∗ = a31 (s) + [µ2 a∗32 (s) + µ1 a∗33 (s)]P2∗ (s). (2.27) s =
By matrix theory, the characteristic roots s i , i=1, 2, 3 of A given in (2.21) are all real and distinct. Defining s0 = 0, it can be shown by partial fraction decompositions that 3 X (sm + λ + µ1 + γ)(sm + λ + µ2 + γ)(sm + γ)2 (s + γ)2 ∗ a11 (s) = 1 + = 1 + c∗11 (s) 3 s Π (s − s )(s − s ) m i m i=1,i6 = m m=0
3 X λp(sm + λ + µ2 + γ)(sm + γ)2 (s + γ)2 ∗ = c∗21 (s) a21 (s) = 3 (s − s )(s − s ) s Π m i m i=1,i6 = m m=0
3 X (s + γ)2 ∗ λ(1 − p)(sm + λ + µ1 + γ)(sm + γ)2 = c∗31 (s) a31 (s) = 3 s (s − s )(s − s ) Π m i m i=1,i6 = m m=0
(s +
γ)a∗12 (s)
=
3 X (sm + γ)µ1 (sm + λ + µ2 + γ)
m=1
(s + γ)a∗22 (s) = 1 +
Π3i=1,i6=m (sm
3 X
− si )(s − sm )
= c∗12 (s)
(sm + γ)[(sm + λ + γ)(sm + λ + µ2 + γ) − λµ2 (1 − p)] Π3i=1,i6=m (sm − si )(s − sm ) m=1
= 1 + c∗22 (s) (s + γ)a∗32 (s) = (s + γ)a∗13 (s) =
3 X
(sm + γ)λµ1 (1 − p) = c∗32 (s) 3 (s − s )(s − s ) Π i m m=1 i=1,i6=m m 3 X (sm + γ)µ2 (sm + λ + µ1 + γ) = c∗23 (s) 3 (s − s )(s − s ) Π i m i=1,i6=m m m=1
(s + γ)a∗33 (s) = 1 +
3 X (sm + γ)[(sm + λ + γ)(sm + λ + µ1 + γ) − λµ1 p]
m=1
Π3i=1,i6=m (sm − si )(s − sm )
= 1 + c∗33 (s)
72
Information and Management Sciences, Vol. 18, No. 1, March, 2007
where c∗ij (s)0 s, denote the summation terms in the above expressions. Using these in (2.22), after some algebraic manipulations, we get P2∗ (s) =
1+
so that P2∗ (s)
=
"
2 α
i ih √ P3 γ 2 w− w 2 −α2 ∗ (s) − c j=1 j1 α α s i h √ 2 2ih P P3 3 w− w −α ∗ ∗ (s) µ c (s) + µ c 2 2 1 j=1 j2 j=1 j3 α2 h
# √ " w− w2 −α2 γ ∗ −h1 (s) 1+2 α s
! #−1 √ w− w2 −α2 ∗ ∗ [µ2 h2 (s)+µ1 h3 (s)] α2 (2.28)
where h∗i (s) =
3 X
c∗ji (s), i = 1, 2, 3.
j=1
The above can be expressed as ∞ X
P2∗ (s) =
(−1)n
n=0
2n+1 αn+1
γ s
!n+1 √ w − w 2 − α2 − h∗1 (s) α
!n+1 √ 2 2 w− w −α α
[µ2 h∗2 (s) + µ1 h∗3 (s)]n
which yields P2∗ (s) =
∞ X
(−1)n
n=0
n+1 X n
2 α
n
µk µn−k (h∗ (s))k (h∗ (s))n−k 2 1 2 3
k h in+1 in+1 √ √ 2 − α2 2 − α2 w − w − w w γ − h∗1 (s) . s αn+1 αn+1 k=0
h
On inversion, we get an explicit expression for P 2 (t) as P2 (t) = γ
∞ X
(−1)
n=0
n
n+1
2 α
(n+1)
Z tX n
+
(−1)n+1
n=0
Z
v 0
n+1
(n−k)
h3
2 α
n
µk µn−k 2 1
k
0 k=0
× ∞ X
(n + 1)
Z
Z
0
uZ v 0
(k)
(n−k)
h2 (y)h3
(v − y)dy
In+1 (α(u − v)) −(λ+µ+γ)(u−v) e dvdu (u − v) n tX
0 k=0
(v − y)h1 (y)dydv
n
µk µn−k
k
2 1
Z
u 0
(k)
h2 (u − v)
In+1 (α(t − u)) −(λ+µ+γ)(t−u) e du (t − u)
(2.29)
Transient Solution of an M/M/2 Queue with Heterogeneous Servers Subject to Catastrophes
73
(n)
where hi (t) is the n-fold convolution of hi (t) with itself. We note that hi (0) = δ(t), the Dirac delta function. Inverting (2.25) - (2.27), after some algebraic manipulation, we get, P0,0 (t) = a11 (t) + γ P1,0 (t) = a21 (t) + γ P0,1 (t) = a31 (t) + γ
Z
Z
Z
t
a11 (u)du +
0 t
a21 (u)du +
0 t 0
a31 (u)du +
Z
Z
t 0
Z
[µ2 a12 (t − u) + µ1 a13 (t − u)] P2 (u)du
(2.30)
[µ2 a22 (t − u) + µ1 a23 (t − u)] P2 (u)du
(2.31)
t 0 t 0
[µ2 a32 (t − u) + µ1 a33 (t − u)] P2 (u)du. (2.32)
Thus (2.12) and (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 and disasters. Theorem 3.1. The steady state distribution of the M/M/2 queue with heterogeneous service rates and catastrophes is obtained as follows: (i) For γ > 0 and λ 6= µ, then
γλ2 [(µ2 − µ1 )p + (λ + µ1 + γ)] P2 = 2 2 γ (λ + µ1 + γ)(λ + µ2 + γ)µ − λµ2 (1 − p) + λµ1 µ2 − µ1 λp
Pn+2 =
h
+ 21 (λ + γ − µ) +
p
i
(λ + µ + γ)2 − α2 [(λ + γ)(λ + µ1 + γ)
(λ + µ + γ) − λµ p(λ + µ + γ) − λµ (1 − p)(λ + µ + γ)] 2 1 2 2 1 n n q
β α
(λ + µ + γ) −
(λ + µ + γ)2 − α2
P2 , n = 1, 2, 3, . . .
(3.1)
γ(λ + µ1 + γ)(λ + µ2 + γ) + µ1 µ2 (2(λ + γ) + µ)P2 (λ+γ)(λ+µ1 +γ)(λ+µ2 +γ)−λµ1 p(λ+µ2 +γ)−λµ2 (1−p)(λ+µ1 +γ) γλp(λ+µ2 +γ)+{µ2 [(λ+γ)(λ+µ2 +γ)−λµ2 (1−p)]+µ1 µ2 λp} P2 P1,0 = (λ+γ)(λ+µ1 +γ)(λ+µ2 +γ)−λµ1 p(λ+µ2 +γ)−λµ2 (1−p)(λ+µ1 +γ) and γλ(1−p)(λ+µ1 +γ)+{µ1 µ2 λ(1−p)+µ1 [(λ+γ)(λ+µ1 +γ)−λpµ1 ]} P2 P0,1 = . (λ+γ)(λ+µ1 +γ)(λ+µ2 +γ)−λµ1 p(λ+µ2 +γ)−λµ2 (1−p)(λ+µ1 +γ) P0,0 =
(3.2) (3.3) (3.4) (3.5)
(ii) For γ > 0 and λ = µ, then γµ2 [(µ2 − µ1 )p + (µ + µ1 + γ)] P2 = 2 2 γ (µ + µ1 + γ)(µ + µ2 + γ)µ − µµ2 (1 − p) + µµ1 µ2 − µ1 µp
h
+ 21 γ +
p
i
(2µ + γ)2 − 4µ2 [(µ + γ)(µ + µ1 + γ)(µ + µ2 + γ)
−µµ p(µ + µ + γ) − µµ (1 − p)(µ + µ + γ)] 1 2 2 1
(3.6)
74
Information and Management Sciences, Vol. 18, No. 1, March, 2007
Pn+2 =
1 2µ
n
(2µ + γ) −
q
(2µ +
γ)2
−
4µ2
n
P2 , n = 1, 2, 3, . . .
(3.7)
γ(µ + µ1 + γ)(µ + µ2 + γ) + µ1 µ2 (3µ + 2γ))P2 (3.8) (µ+γ)(µ+µ1 +γ)(µ+µ2 +γ) − µµ1 p(µ+µ2 +γ) − µµ2 (1 − p)(µ+µ1 +γ) γµp(µ+µ2 +γ)+{µ2 [(µ+γ)(µ+µ2 +γ)−µµ2 (1−p)]+µ1 µ2 µp} P2 P1,0 = (3.9) (µ+γ)(µ+µ1 +γ)(µ+µ2 +γ) − µµ1 p(µ+µ2 +γ)−µµ2 (1−p)(µ+µ1 +γ) and γµ(1 − p)(µ+µ1 +γ) + {µ1 µ2 µ(1 − p)+µ1 [(µ+γ)(µ+µ1 +γ)−µpµ1 ]} P2 P0,1 = . (3.10) (µ+γ)(µ+µ1 +γ)(µ+µ2 +γ)−µµ1 p(µ+µ2 +γ)−µµ2 (1−p)(µ+µ1 +γ) P0,0 =
Proof. For γ > 0 and λ 6= µ, from (2.24), we have P2∗ (s) =
n
h
s (s + γ) µ2
P3
h
(s + γ) 1 − (s + γ)
∗ j=1 aj2 (s)+µ1
P3
P3
∗ j=1 aj1 (s)
∗ j=1 aj3 (s)
i
i
h io . √ +(s+λ+γ)− 21 w− w2 −α2
Multiplying the above equation by s on both sides and taking limit as s → 0, we get γλ2 [(µ2 − µ1 )p + (λ + µ1 + γ)] lim sP2∗ (s) = s→0 2 2 γ (λ + µ1 + γ)(λ + µ2 + γ)µ − λµ2 (1 − p) + λµ1 µ2 − µ1 λp
h
+ 21 (λ + γ − µ) +
p
i
(λ + µ + γ)2 − α2 [(λ + γ)(λ + µ1 + γ)
. (3.11)
(λ + µ + γ) − λµ p(λ + µ + γ) − λµ (1 − p)(λ + µ + γ)] 2 1 2 2 1
The result (3.1) follows directly from (3.11), by using Tauberian theorem. Taking Laplace transform of (2.12), we have ∗ Pn+2 (s)
=
n
β α
(s+λ+µ+γ)−
q
(s+λ+µ+γ)2 −α2
n
P2∗ (s), n = 1, 2, 3, . . . . (3.12)
As before, multiplying (3.12) by s on both sides and taking limit as s → 0, we get ∗ lim sPn+2 (s) = lim
s→0
s→0
n
β α
(s + λ + µ + γ) −
q
(s + λ + µ + γ)2 − α2
n
n = 1, 2, 3, . . .
sP2∗ (s), (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 > 0 and λ = µ, the results (3.6) - (3.10) 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 and only if γ > 0 or γ = 0 and λ < µ 1 + µ2 . 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 + µ2 . It is interesting to note that, for γ = 0 and λ < µ1 + µ2 , our results (3.1) - (3.5) agree with Rubinovitch’s results.
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 and 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 and 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)) = P1,0 (t) + P0,1 (t) +
∞ X
(n + 2)Pn+2 (t).
n=0
Using (2.12), (2.31) and (2.32), the above can be written as E(X(t)) = a21 (t) + γ
Z
t 0
+a31 (t) + γ +2P2 (t) +
Z
a21 (u)du +
0 ∞ X
t
Z
t 0
a31 (u)du + (n + 1)nβ n
n=1
[µ2 a22 (t − u) + µ1 a23 (t − u)]P2 (u)du
Z
Z
0 t
0
t
[µ2 a32 (t − u) + µ1 a33 (t − u)]P2 (u)du P2 (u)e−(λ+µ+γ)(t−u)
In (α(t − u)) du (t − u)
(4.1)
where P2 (t) is given in (2.29). If γ > 0, the mean number of customers in the system under steady state is computed as p
2µ{4µ − [(λ + µ + γ) − (λ + µ + γ)2 − 4λµ]}P2 p E(X) = {2µ − [(λ + µ + γ) − (λ + µ + γ)2 − 4λµ]}2
γ[λp(µ − µ ) + λ(λ + µ + γ)] + P {µ (γ + λ)[(λ + µ + γ) 2 1 1 2 2 2 +µ λ − λ(1 − p)] + µ [(λ + γ)(λ + µ + γ) − µ λp]} 1 1 1 1 , + (λ + γ)(λ + µ + γ)(λ + µ + γ) − λµ p(λ + µ + γ) 1
−λµ (1 − p)(λ + µ + γ) 2 1
and
2
1
2
if λ 6= µ (4.2)
76
Information and Management Sciences, Vol. 18, No. 1, March, 2007
p
2µ{2µ − γ + 4µγ + γ 2 }P2 p E(X) = { 4µγ + γ 2 − γ}2
γ[µp(µ − µ ) + µ(µ + µ + γ)] + P {µ (γ + µ)[(µ + µ + γ) 2 1 1 2 2 2 +µ µ − µ(1 − p)] + µ [(µ + γ)(µ + µ + γ) − µ µp]} 1 1 1 1 , + (µ + γ)(µ + µ + γ)(µ + µ + γ) − µµ p(µ + µ + γ) 1
2
1
2
−µµ (1 − p)(µ + µ + γ) 2 1
if λ = µ (4.3)
where P2 is given in (3.1) for λ 6= µ and in (3.6) for λ = µ. Probability of arriving customers joining the queue: The probability that an arriving customer is required to join the queue at time t is given by P (X(t) ≥ 2) =
∞ X
Pn+2 (t)
n=0
= P2 (t) +
∞ X
nβ n
n=1
Z
t 0
P2 (u)e−(λ+µ+γ)(t−u)
In (α(t − u)) du. (t − u)
(4.4)
Similarly, for γ > 0, the steady state probability that an arriving customer joins the queue is P (X ≥ 2) =
∞ X
Pn+2 =
n=0
The number of busy servers:
√2µP2 (µ−λ−γ)+ (λ+µ+γ)2 −4λµ √ 22µP2 γ +4γµ−γ
, if λ 6= µ , if λ = µ.
(4.5)
Let M (t) denote the number of busy servers at time t. The probability that the system has n busy servers is given as, P (X(t) = 1) = P (t) + P (t) , for n = 1 1,0 0,1 P {M (t) = n} = P ∞ P (X(t) > 1) = , for n = 2 n=0 Pn+2 (t)
(4.6)
and the corresponding steady state probability is obtained for γ > 0 and λ 6= µ as
P (M = n) =
γ[λp(µ2 −µ1 )+λ(µ1 +λ+γ)]+P2 {µ2 [(λ+γ)(λ+µ2 +γ) −λ(1−p)+µ1 λ]+µ1 [(λ+γ)(λ+µ1 +γ)−µ1 λp]} , if n = 1
[(λ + γ)(µ1 + λ + γ)(λ + µ2 + γ) − λµ1 p(λ + µ2 + γ)
−λµ2 (1 − p)(λ + µ1 + γ)] √2µP2 {µ−λ−γ+
(λ+µ+γ)2 −4λµ}
(4.7)
, if n = 2.
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 > 0 and 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)) = P1,0 (t) + P0,1 (t) + 2
∞ X
Pn+2 (t)
n=0
which can be simplified as E(M (t)) = 2[1 − P0,0 (t)] − [P1,0 (t) + P0,1 (t)].
(4.8)
For γ > 0, the corresponding steady state result is given as 2[λ(λ + µ + γ)(λ + µ + γ)−λµ p(λ + µ + γ)−λµ (1 − p)(λ + µ + γ) 1 2 1 2 2 1
−µ µ (2(λ + γ) + µ)]P −γ[λp(µ − µ ) + λ(λ + µ + γ)]]+{[µ ((λ + γ)
1 2 2 2 1 1 2 (λ + µ + γ)) − λ(1 − p) + µ λ] + µ [(λ + γ)(λ + µ + γ) − µ λp]}P 2 1 1 1 1 2 E(M ) = (λ + γ)(λ + µ + γ)(λ + µ + γ) − λµ p(λ + µ + γ) 1
2
1
2
−λ µ (1 − p)(λ + µ + γ) 2 1
,
if λ 6= µ (4.9)
and 2[µ(µ + µ + γ)(µ + µ + γ)−µµ p(µ + µ + γ)−µµ (1 − p)(µ + µ + γ) 1 2 1 2 2 1
−µ µ ((3µ + 2γ))]P − γ[µp(µ − µ ) + µ(µ + µ + γ)]] + {[µ ((µ + γ)
1 2 2 2 1 1 2 (µ + µ + γ)) − µ(1 − p) + µ µ] + µ [(µ + γ)(µ + µ + γ) − µ µp]}P 2 1 1 1 1 2 E(M ) = (µ + γ)(µ + µ + γ)(µ + µ + γ) − µµ p(µ + µ + γ) 1
2
1
−µ µ (1 − p)(µ + µ + γ) 2 1
2
,
if λ = µ. (4.10)
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 and 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. P0,0 = lim P (X(t) = 0) = t→∞
1 λ
E(T0 ) + E(L)
78
Information and Management Sciences, Vol. 18, No. 1, March, 2007
where T0 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 E(T0 ) =
1 λ
so that E(L) =
1 −1 (P − 1), λ 0,0
which yields E(L) =
λ(λ + γ) {(1 − p)(µ1 − µ2 ) + (λ + µ2 + γ)} − µ1 µ2 (2(λ + γ) + µ)P2 , λγ(λ + µ1 + γ)(λ + µ2 + γ) + λµ1 µ2 (2(λ + γ) + µ)P2 if λ 6= µ (4.11)
and E(L) =
µ(µ + γ) {(1 − p)(µ1 − µ2 ) + (µ + µ2 + γ)} − µ1 µ2 (2γ + 3µ)P2 , µγ(µ + µ1 + γ)(µ + µ2 + γ) + µµ1 µ2 (3µ + 2γ)P2 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
E(N ) =
(λ + γ)(λ + µ + γ)(λ + µ + γ) − λµ p(λ + µ + γ) 1 2 1 2 −λµ (1 − p)(λ + µ + γ) 2
1
γ(λ + µ1 + γ)(λ + µ2 + γ) + µ1 µ2 (2(λ + γ) + µ)P2
, if λ 6= µ
(4.13)
, if λ = µ.
(4.14)
and
E(N ) =
(µ + γ)(µ + µ + γ)(µ + µ + γ) − µµ p(µ + µ + γ) 1 2 1 2 −µµ (1 − p)(µ + µ + γ) 2
1
γ(µ + µ1 + γ)(µ + µ2 + γ) + µ1 µ2 (2γ + 3µ)P2
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
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 and 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, 233-249, 2000. [2] C. Baburaj, On the transient distribution of a single and batch service queueing system with accessibility to the batches, International Journal of Information and Management Sciences, Vol. 11, No. 2, 27-36, 2000. [3] R. J. Boucherie and O. J. Boxma, The workload in the M/G/1 queue with work removal, Probability in Engineering and Informational Sciences, Vol. 10, 261-277, 1996. [4] X. Chao, M. Miyazawa and M. Pinedo, Queueing Networks, Customers, Signals and Product form Solutions, John Wiley and Sons, Chichester, 1999. [5] V. D¨ orrsam, Materialfluβorientierte Leistungsanalyse einstufiger Produktionssysteme, Dissertation, Universit¨ at Karlsruhe (in German), 1999. [6] A. N. Dudin and S. Nishimura, A BMAP/SM/1 queueing system with Markovian arrival input of disasters, Journal of Applied Probability, Vol.36, 868-881, 1999. [7] P. C. Garg, A measure for time dependent queueing problem with service in batches of variable size, International Journal of Information and Management Sciences, Vol. 14, No. 4, 83-87, 2003. [8] E. Gelenbe and G. Pujolle, Introduction to Queueing Networks, (2nd edition), John Wiley and Sons, Chichester, 1998. [9] G. Jain and K. Sigman, A Pollaczek-Khinchine formula for M/G/1 queues with disasters , Journal of Applied Probability, Vol. 33, 1191-1200, 1996. [10] S. S. Lavenberg, A perspective on queueing models of computer performance in queueing theory and its applications, Liber Amicorum for J. N. Cohen; CWI Monograph 7, 1988. [11] E. D. Lazowaska, G. S. Zahorjan and K. C. Sevcik, Quantitative system performance, Prentice-Hall, Englewood Cliffs, 1984 . [12] W. Liu and P. Kumar, Optimal control of a queueing system with two heterogeneous servers, IEEE Transactions on Automatic Contol, Vol. 29, 696-703, 1984. [13] M. Mittler, The variability of cycle times in semiconductor manufacturing, Dissertation, Universit¨ at W¨ urzburg, 1997. [14] M. Rubinovitch, The slow server problem, Journal of Applied Probability, Vol. 22, 205-213, 1985. [15] V. P. Singh, Two-server Markovian queues with balking: Heterogeneous vs. homogeneous servers, Operations Research, Vol.19, 145-159, 1970. [16] G. N. Watson, A Treaties on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1962. [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, 395-408, 1983.
80
Information and Management Sciences, Vol. 18, No. 1, March, 2007
Authors’ Information B. Krishna Kumar is an Assistant Professor in the Department of Mathematics, Anna University, India. He received his M.Sc. and M.Phil degrees from University of Madras in 1984 and 1985 respectively. Subsequently, he has obtained his Ph.D degree from I. I. T., Madras in 1991. He had been in N.T.T., Multimedia Laboratory, Tokyo as a post-doctoral fellow during 1993-1994. His research interests include Queueing Models and Their Applications, Branching Processes and Mathematical Ecology. Dr. B. Krishna Kumar is a member of the Indian Society for Probability and Statistics and the Operations Research Society of India. Department of Mathematics, Anna University, Chennai, India. E-mail:
[email protected]
TEL : +91-44-22203308.
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 and Stochastic Modelling of Communication systems. Department of Mathematics, R. M. K. Engineering College, Kavaraipettai, Thiruvallur Dt., India. E-mail: pavai
[email protected]
TEL : +91-44-26563353.
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 and Stochastic Dynamic Systems. He is a member of the Operations Research Society of India. Department of Mathematics, Anna University, Chennai, India.
