UNRELIABLE M X /G/1 QUEUE WITH MULTI-OPTIONAL SERVICES AND MULTI-OPTIONAL VACATIONS UNDER N-POLICY Madhu Jain Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Haridwar-247667, India E-mail:
[email protected]
A multiple optional vacation policy for the unreliable server batch arrival queueing system with essential and multi optional services under N-policy is investigated. According to vacation policy, the server takes an essential phase of vacation after the completion of service of an individual customer when queue becomes empty. On returning from the vacation, if the server finds the queue empty, then he again goes for second phase of vacation with some probability. This process is continued for at most ’m’ phases of vacations. The server follows N-policy according to which the server turns on and starts the service only when the queue size reaches or exceeds N. Once the queue size reaches N, the server provides ”essential” service to all the arriving units whereas ”multioptional” services to only a few of them who demand for the same. By employing generating function and supplementary variable techniques, the system size distribution and other performance indices such as average queue length and long run probabilities of the system states have been derived. The effects of the system parameters on various performance measures are examined by taking an illustration. Keywords : Unreliable server, Bulk queue, N-policy, Essential service, Optional multi-phase services, Optional multi-phase vacations, Generating function, Supplementary variables, Queue length.
1. Introduction Vacation models depict a very important class of real life congestion situations encountered in day-to-day as well as industrial scenario. The vacation in queueing context can be considered as the period during which the server
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is not available as he has left when the system becomes empty. Some times, the server remains in idle state until queue size reaches to a threshold level. This type of queueing models are studied in queueing literature under N policy. Whenever the server is in working state, he may break down at any time, and if the server fails, it is immediately sent for repair. In many queueing problems, server unreliability makes negative impact on the performance of any system and therefore some measures are to be taken to maintain a desired grade of service. The queueing systems with unreliable server reflect more practical situations that we face in real time systems. Gray (1) investigated a queueing model with multiple types of breakdown. Wang et al.(2) have done the maximum entropy analysis to the N-policy M/G/1 queueing system with server breakdown and general startup. Other denominations of such queueing systems can be seen from the works of Jain and Agarwal (3), Ke and Lin (4), Choudhury et al.(5), Ke and Chang (6) and many others. In many service systems, the service is provided by the server in more than one phase to improve the quality of service and to fulfill customer’s satisfaction. Such type of queueing models have been studied by few researchers. Choudhury and Madan (7) and Choudhury and Paul (8) considered a batch arrival queueing system wherein the server renders two stages of heterogeneous service with Bernoulli vacation schedule under N-policy. The latter have demonstrated the existence of stochastic decomposition property to show that the departure point queue size distribution of this model can be decomposed into the distribution of three independent variables. The extensive stationary analysis of an M X /G/1 queue with two phase service under multiple vacation policy has been carried out by Chaudhury et al. (9). The steady state analysis of a queueing system with two phases of heterogeneous service and Bernoulli vacation schedule was considered by Choudhury (10). Further, Wang and Li (11) and Kumar and Arumuganathan (12) developed a retrial queueing model wherein the server provides two phases of service in succession. They have suggested the application of the proposed model to the analysis of a communication protocol. Though the provision of providing more than one phases of service by the server improves the grade of service but it increases the service time which in turn increases the system cost and customer’s discouragement. In view of this, efforts have been made by the many researchers to develop queueing models wherein the server provides first essential service to all the arriving customers
3
but one or more than one optional services to only a few of them who demand for the same. The contributions to such queueing models can be seen in the works of Madan (13), Medhi (14) and Choudhury (15) and many others. A batch arrival queue with a second optional service channel under N-policy was investigated by Choudhary and Madhuchanda (16) and Choudhury and Paul (17). Madan and Al-Rawwash (18) has incorporated the concept of optional vacations for the server preceded by the first essential phase of vacation to analyze M X /G/1 queue. Ke (19) examined an M X /G/1 queueing system with startup server and J additional options for service. Recently, an optimal repairable M X /G/1 queue with multi-optional services and Bernoulli vacation has been discussed by Jain and Upadhyaya (20).
2. Model Description Consider a single server queueing system in which batches of customers arrive according to a Poisson process with rate λ . The customers arrive in the system in a batch of size k with probability ck . To formulate the mathematical model, we consider general i.i.d. distributed random variables for different phases of service time, vacation time and repair time which are described as follows: (A) Service Process: A single server provides a preliminary first essential service (F ES), denoted by B1 , to all arriving customers. As soon as the F ES of a customer is completed, the customer may be provided with a second optional service (SOS), denoted by B2 , with probability q1 (0 ≤ q1 ≤ 1) if he demands for the same, or may leave the system with probability q1 (= 1 − q1 ). After the completion of SOS, the customer may demand a third optional service (T OS), denoted by B3 , with probability q2 (0 ≤ q2 ≤ 1), or may leave the system with probability q¯2 (= 1−q2 ). In general, the customer may opt any of the ith (2 ≤ i ≤ l) optional services, denoted by Bi , with probability qi−1 , or may leave the system with probability q¯i−1 (= 1 − qi−1 ). The service discipline is assumed to be F CF S (First Come First Served). Further, it is assumed that the service time random variable Bi (1 ≤ i ≤ l) follows the general probability law with cumulative distribution function cdf
4
Bi (u), Laplace Steiltjes transforms LST and finite moments E[Sir ], r ≤ 1, (1 ≤ i ≤ l). (B) Vacation Schedule: Once F ES and ith (1 ≤ i ≤ l) optional phase service of a customer are completed, the server may go for the first essential phase of vacation of random length V. As soon as the first essential phase of vacation of the server is completed, he may either opt for second optional vacation, with probability θ1 (0 ≤ θ1 ≤ 1) or may remain with the system in idle state until customer arrives. Continuing this process, the server can take at most i (1 ≤ i ≤ m) optional phases of vacations with probability i (2 ≤ i ≤ m). Furthermore, we assume that the vacation time random variable follows a general law of probability with cdf Fi (u), LST Fi∗ (s) and finite moments E[Vir ], r ≥ 1 (1 ≤ i ≤ m) and is independent of service time random variables Bi (1 ≤ i ≤ l) and arrival process. (C) Repair Process: During the busy state, the server is subject to breakdown while rendering ith phase service according to Poisson distribution with breakdown rate αi ; (1 ≤ i ≤ l) the broken down server is repaired with repair rate ηi (v) (1 ≤ i ≤ l) to restore the service. We assume that while the server is broken down during ith phase service, the random variable Gi (1 ≤ i ≤ l) denotes the repair time which follows a general law of probability with cdf Gi(v), LST G∗i (s) , (1 ≤ i ≤ l) and finite moments E[Rir ], r ≥ 1. We define the hazard rates when the server is busy with ith phase service, on ith phase vacation state and under repair state while broken down during the ith phase service, respectively by
µi (u) =
βi (u) fi (u) gi (v) , 1 ≤ i ≤ l; ξi (u) = ¯ ; 1 ≤ i ≤ m; ηi (v) = ¯ , 1 ≤ i ≤ l. Bi¯(u) Fi (u) Gi (v)
where bar is used for complementary distribution i.e. complementary ¯ = 1 − A(u). distribution function of A(u) is A(u) The probabilities of different states of the system are defined as follows: Dn (t) : Probability that the server is idle at time t and there are n customers in the system.
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P1,n (t, u) : Probability that there are n customers in the system at time t and the server is busy in providing FES to the customer and the customer is being served with elapsed service time lying between u and u+du. Pi,n (t, u) : Probability that there are n customers in the system at time t and the server is busy in providing ith (2 ≤ i ≤ l) optional service to the customer and the customer is being served with elapsed service time lying between u and u+du. Q1,n (t, u) : Probability that there are n customers in the system at time t and the server is in first essential phase of vacation with elapsed vacation time lying between u and u+du. Qi,n (t, u) : Probability that there are n customers in the system at time t and the server is on ith (2 ≤ i ≤ m) optional phase of vacation with elapsed vacation time lying between u and u+du. Ri,n (t, u) : Joint probability that there are n customers in the system at time t, the elapsed service time of the customer undergoing FES is equal to u and the server is undergoing the repair with elapsed repair time lying between v and v+dv . Ri,n (t, u) : Joint probability that there are n customers in the system at time t, the elapsed service time of the customer undergoing ith (2 ≤ i ≤ l) optional service is equal to u and the server is undergoing the repair with elapsed repair time lying between v and v+dv .
3. Queue Size Distribution At A Random Epoch In this section, we employ the supplementary variable technique (SVT) to convert non-markovian M [x] /G/1 queue into markovian one by introducing supplementary variables corresponding to elapsed service time, repair time and vacation time. Let Nq (t) denotes the number of customers in the queue at time t. The state of the system at time t can be described by stochastic process X(t) = (C(t), Nq (t), ζ(t)) where C(t) is equal to 0, 1 or i according to whether the server is idle, busy with FES or busy with ith (2 ≤ i ≤ l) optional service, at time t; Nq (t) denotes the number of customers in the queue at time t. If C(t) ∈ {1, 2, ..., l}, then ζ(t) represents the corresponding elapsed time of service in progress.
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We define the limiting probabilities at the steady state of system as: Dn = lim Dn (t); n ≥ 0;
Pi,n (u) = lim Pi,n (t, u), n ≥ 1,
t→∞
t→∞
Ri,n (u, v) = lim Ri,n (t, u, v), 1 ≤ i ≤ l;
Qi,n (u) = lim Qi,n (t, u), n ≥ 1, 1 ≤ i ≤ m.
t→∞
t→∞
The Chapman-Kolmogorov equations governing the model are: Z∞ λD0 =
∞
ξm (u)Qm,0 (u)du +
l−1 Z X
q¯i µi (u)Pi,1 (u)du
(1)
i=1 0
0
λD0 = λ
n X
ck Dn−k ; 1 ≤ n ≤ N − 1
(2)
k=1 n
[
X ∂ + µ1 (u) + λ + α1 ]P1,n (u) = λ ck P1,n−k (u) + ∂u k=1
Z∞ η1 (v)R1,n (u, v)dv; n ≥ 1 0
(3) n
X ∂ [ + µi (u) + λ + αi ]Pi,n (u) = λ ck Pi,n−k (u) + ∂u k=1
Z∞ ηi (v)Ri,n (u, v)dv (4) 0
n
[
X ∂ + ξi (u) + λ]Qi,n (u) = λ ck Qi,n−k (u); n ≥ 1; 1 ≤ i ≤ m ∂u
(5)
∂ + ξi (u) + λ]Qi,0 (u) = 0; 1 ≤ i ≤ m ∂u
(6)
k=1
[
n
[
X ∂ + ηi (v) + λ]Ri,n (u, v) = λ ck Ri,n−k (u, v); n ≥ 1; 1 ≤ i ≤ l ∂v
(7)
k=1
The boundary conditions under steady state are P1,n (0) =
l−1 X i=1
P1,n (0) =
l−1 X i=1
Z∞ q¯i
Z∞ µi (u)Pi,n+1 (u)du +
0
Z∞ q¯i
(8)
0
Z∞ µi (u)Pi,n+1 (u)du+
0
ξm (u)Qm,n (u)du
ξm (u)Qm,n (u)du+λ 0
n X
ck Dn−k ; n ≥ N
k=1
(9)
7
Z∞ µi−1 (u)Pi−1,n (u)du; n ≥ 0; 2 ≤ i ≤ l
Pi,n (0) = qi
(10)
0
Z∞ µl (u)Pl,n+1 (u)du; n ≥ 0
Q1,n (0) =
(11)
0
Z∞ ξi−1 (u)Qi−1,n (u)du; n ≥ 0; 2 ≤ i ≤ m
Qi,n (0) = θi−1
(12)
0
Ri,n (u, 0) = αi Pi,n (u); n ≥ 1; 1 ≤ i ≤ l
(13)
We define probability generating functions as follows:
Pi (u, z) =
∞ X
Pi,n (u)z n , 1 ≤ i ≤ l; Qi (u, z) =
n=1
∞ X
Qi,n (u)z n , 1 ≤ i ≤ m;
n=1
Ri (u, v, z) =
∞ X
Ri,n (u, v)z n , 1 ≤ i ≤ l; C(z) =
n=1
∞ X
Cx z x
x=1
The normalizing condition is N −1 X n=0
Dn +
m ∞ X X
Qi,n (u)du +
n=0 i=1
Z∞ Z∞ l Z∞ ∞ X X [ Pi,n (u)du + Ri,n (u, v)dudv] = 1 n=1 i=1 0
0
0
(14) Denote ¯ C(z) = 1 − C(z) Theorem 1: The partial probability generating functions at random epochs when the server is busy with FES, busy with ith (2 ≤ i ≤ l) optional service, on ith (1 ≤ i ≤ m) optional vacation and under repair while broken down during ith (1 ≤ i ≤ l) phase service respectively, are given by P1 (u, z) =
¯ [−λzD(z)C(z) exp{−h1 (z)u}B¯1 (u)] A(z)
(15)
8
¯ [−λzD(z)C(z)
i−1 Q
qj βj ∗ (hj (z))exp{−hi (z)u}B¯i (u)]
j=1
Pi (u, z) =
;2 ≤ i ≤ l
A(z)
(16)
∗ ¯ [−λD(z)C(z)β l (hl (z))
l−1 Q
¯ qj βj ∗ (hj (z))(exp{−λC(z)u} F¯i (u)]
j=1
Q1 (u, z) =
A(z) (17)
¯ [−λD(z)C(z)
l−1 Q
qj βj ∗ (hj (z))βl ∗ (hl (z))
j=1
Qi (u, z) =
i−1 Q
¯ ¯ θj fj ∗ (λC(z))(exp{−λ C(z)u} F¯i (u)]
j=1
;
A(z) (18) 2≤i≤m
R1 (u, v, z) =
¯ ¯ [−λα1 zD(z)C(z) exp{−hi (z)u − λC(z)v} B¯1 (u)G¯1 (v)] A(z)
(19)
i−1 Q ¯ ¯ [−λα1 zD(z)C(z) qj βj ∗ (hj (z)) exp{−hi (z)u − λC(z)v} B¯i (u)G¯i (v)] j=1
Ri (u, v, z) =
A(z) (20) 2≤i≤l
where ¯ ¯ hi (z) = [λC(z) + αi (1 − gi∗ (λC(z)))]f or1 ≤ i ≤ l A(z) = z−¯ q1 β1 ∗ (h1 (z))−
l−1 X
q¯i βi ∗ (hi (z))
i=2
M (z) = βi ∗ (hi (z))
l−1 Y
qj βj ∗ (hj (z))−M (z)
j=1 l−1 Y j=1
i−1 Y j=1
qj βj ∗ (hj (z))
¯ θj fj ∗ (λC(z))−M (z)
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Proof: Multiplying equations (1)-(13) by the appropriate powers of z and summing over n = 0, 1, 2, 3.... and then solving, we get the required results. Theorem 2: The marginal generating function for the idle state at random epoch is obtained by D(z) = ρ¯H(z) (21) where, ρ = [λC 0 (1)[E[S1 ] +
l X
qi−1 E[Si ] + E[V1 ] +
i=2
θi−1 E[Vi ]]];
i=2 NP −1
H(z) =
m X
φj z j
j=0 NP −1
φj
j=0
φn = P r(a batch of customers finds at least ’n’ units in the system during an idle period) and NP −1 φj is the mean number of batches arriving during an idle period. j=0
Proof: Multiplying equations (1)-(13) by the appropriate powers of Z and summing over n = 0, 1, 2, 3.... and then solving, we get the required results.
Theorem 3: At random epoch, the marginal generating function for the FES state, ith (2 ≤ i ≤ l) optional service state, ith (1 ≤ i ≤ l) optional vacation state and repair state while broken down during ith (1 ≤ i ≤ l) phase service, respectively are obtained as P1 (z) =
¯ β¯∗ (h1 (z))] [−λzD(z)C(z) 1 (h1 (z))A(z)
¯ β¯∗ (hi (z)) [−λzD(z)C(z) i Pi (z) =
qj βj ∗ (hj (z))]
j=1
(hi (z))A(z) [−D(z)
Q1 (z) =
i−1 Q
l−1 Q
(22)
;2 ≤ i ≤ l
(23)
¯ qj βj ∗ (hj (z))βl ∗ (hl (z))f1 ∗ (λC(z))]
j=1
A(z)
(24)
10
[−D(z)
l−1 Q
qj βj ∗ (hj (z))βl ∗ (hl (z))
j=1
Qi (z) =
i−1 Q
∗ ¯ ¯ θj fj ∗ (λC(z))f i (λC(z))]
j=1
;2 ≤ i ≤ m
A(z)
(25) ¯ [−α1 zD(z)β1 ∗ (h1 (z))gl ∗ (λC(z))] (h1 (z))A(z)
(26)
¯ [−αi zD(z)βi ∗ (hi (z))gi ∗ (λC(z))] ;2 ≤ i ≤ l (hi (z))A(z)
(27)
R1 (z) = Ri (z) =
Proof: By using Z∞ Qi (z, u)du; 1 ≤ i ≤ m
Qi (z) = 0
Z∞ Pi (z) =
Z∞ Z∞ Ri (u, v, z)dudv; 1 ≤ i ≤ l; 1 ≤ i ≤ l
Pi (z, u)du; Ri (z) = 0
0
0
we obtain results given in (25)-(27), respectively. Theorem 4: The probability generating function for the number of customers in the queue at the random epoch is given by LQ (z) = D(z) +
l X
Pi (z) + z[
m X
Qi (z)] +
Ri (z)
(28)
i=1
i=1
i=1
l X
Proof: The marginal probabilities obtained are summed up to give the probability generating function of queue size distribution at random epoch as LQ (z) = D(z) +
l X i=1
Pi (z) + z[
m X i=1
Qi (z)] +
l X
Ri (z)
i=1
4. Long Run Probabilities And Average Queue Length In this section, we obtain explicit expressions for the system state probabilities and the queue size at random epoch for the queueing model with essential
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service, second optional phase service, essential vacation and second optional phase vacation to investigate the behavior of queueing system under consideration. Theorem 5: The probabilities that the server is in idle state, busy with FES, busy with SOS, on first essential vacation, on second optional phase vacation and under repair during FES and SOS are denoted by P(I), P(S1), P(S2), P(V1), P(V2), P(R1) and P(R2), respectively and are given by P (I) =
N −1 X
Dn = ρ¯
n=0
where, ρ = [λC 0 (1){E[S1 ] + q1 E[S2 ] + θq1 E[V ]}]
P (S1 ) =
P (S2 ) =
(29)
λC 0 (1)D(1)E[S1 ] [¯ ρ − λC 0 1{α1 E[S1 ]E[R1 ] + q1 α2 E[S2 ]E[R2 ]θq1 E[V ]}
[¯ ρ−
λq1 C 0 (1)D(1)E[S2 ] 0 λC 1{α1 E[S1 ]E[R1 ] + q1 α2 E[S2 ]E[R2 ]θq1 E[V
]}
(30)
(31)
λq1 C 0 (1)D(1)E[V ] [¯ ρ − λC 0 1{α1 E[S1 ]E[R1 ] + q1 α2 E[S2 ]E[R2 ]θq1 E[V ]}
(32)
λθ1 q1 C 0 (1)D(1)E[V ] 1 E[S1 ]E[R1 ] + q1 α2 E[S2 ]E[R2 ]θq1 E[V ]}
(33)
P (R1 ) =
λα1 C 0 (1)D(1)E[S1 ]E[R1 ] [¯ ρ − λC 0 1{α1 E[S1 ]E[R1 ] + q1 α2 E[S2 ]E[R2 ]θq1 E[V ]}
(34)
P (R1 ) =
λα2 C 0 (1)D(1)E[S2 ]E[R2 ] [¯ ρ − λC 0 1{α1 E[S1 ]E[R1 ] + q1 α2 E[S2 ]E[R2 ]θq1 E[V ]}
(35)
P (V1 ) =
P (V2 ) =
[¯ ρ−
λC 0 1{α
Proof : To derive above results , we have used P (S1 ) = lim P1 (z); P (S2 ) = lim P2 (z); P (V1 ) = lim Q1 (z); z→1
z→1
z→1
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P (V2 ) = lim Q2 (z); z→1
LQ = lim L0 (z) z→1
Theorem 6: The number of customers in the queue in case of second optional phase service and second optional phase vacation at random epoch is given by P (S1 ) = lim P1 (z) z→1
L(Q) = D(1) +
+ α1
[X2 (Y2 + Z2 + F2 − M ) − G2 ] [X1 (Y1 + Z1 + F1 ) − G1 ] + q1 2 2B1 2B22
[X1 (H1 I1 + J1 − K1 ) − W1 ] [X2 (H2 I2 + J2 − K2 − M ) − W2 ] + q1 α2 2X12 2X22 (36)
+λq1 C 0 (1)
[A0 (1)(N2 + O2 + P2 ) − U2 ] [A0 (1)(N1 + O1 + P1 ) − U1 ] 0 +λθq C (1) 1 2A0 (12 2A0 (22
where for i ∈ {1, 2} Xi = h0i (1)A0 (1); Yi = λD(1)C 0 (1)2h0i (1)E(Si ) − ((h0i )2 )E(Si2 ) Zi = 2λD0 (1)C 0 (1)h0i (1)E(Si ); Fi = λD(1)C 00 (1)h0i (1)E(Si ), Gi = λD(1)C 0 (1)(h0i (1))2 A00 (1)E(Si ); Hi = D(1)h0i (1)E(Si ) : Ii = λC 00 (1)E(Ri ) + λ2 (C 0 (1))2 E(Ri2 ); Ki = λD(1)C 0 (1)(h0i (1))2 E(Si2 )E(Ri ); Li = λD(1)C 0 (1)(h02 (1))2 A00 (1)E(Si )E(Ri ); M = 2λD(1)C 0 (1)h01 (1)h02 (1)E(S1 )E(S2 ); N1 = λD(1)C 0 (1)E(V12 ); O1 = 2D0 (1)E(V1 ); P1 = 2D(1){1 − h01 (1))E(S1 ) − h02 (1)E(S2 )}E(V1 ); U1 = D(1)A00 (1)E(V1 ); N2 = λD(1)C 0 (1)E(V22 ); O2 = 2D0 (1)E(V2 ); U2 = D(1)A00 (1)E(V2 ); P2 = 2D(1){1−h01 (1))E(S1 )−h02 (1)E(S2 )}E(V2 ); h0i (1) = −λC 0 (1){1+αi E(Ri )}; D(1) = 1 − ρ; A0 (1) = 1 + h01 (1))E(S1 ) + q1 h02 (1)E(S2 ) − λrq1 C 0 (1)E(V ); 5. Numerical Illustration
13
Fig. 1(a): Effect of LQ vs
for (a) Mx/
/1
Fig. 2: Effect of LQ vs 1for Mx/ /1
Fig. 3(a): Effect of LQ vs
1 for Mx/ /1
Fig. 4(a): Effect of LQ vs for Mx/ /1
Fig. 1(b) Mx/M/1 models by varying q1
Fig. 2 (b) ; Mx/M/1 models by varying q1
Fig. 3 (b) Mx/M/1 models by varying q1
Fig. 4 (b): Mx/M/1 models by varying q1
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In this section, we are interested in sensitivity analysis by taking numerical illustration. We have developed a computer program by using mathematical software MATLAB. The numerical results are summarized in figs 1-4. For computation purpose, we consider M X /γ/1 and M X /M/1 models. We also assume that batch size distribution is geometric distributed. For the sake of convenience, we set default parameters as p = 0.5, λ = 0.4, µ1 = 2, µ2 = 6, g1 = 0.1 = g2 , ν = 0.2, q1 = 0.2, θ1 = 0.1, α1 = 0.02 = α2 and N = 6,, for figures 1-4. Figs 1-4 show the effect of parameters such as λ , µ1 , ν and α1 on the average queue length by varying optional service probability q1 . These figs depict that there is an increasing trend in the average queue length followed by gradual increments with the increase in the values of arrival rate and breakdown rate α1 . On the other hand, we see a decreasing trend in LQ by varying µ1 and ν . In figs 2-4, we note that the average queue length attains a constant value for higher arrival rate and breakdown rate as well as for lower service rate and vacation rate. Moreover, in figs 1-4, we visualize that LQ increases on increasing q1 , which can be seen in many service systems wherein the service is provided in more than one phases. Overall we conclude that the queue length can be controlled up to a limit by setting appropriate system parameters.Our investigation shows that it is beneficial for the server to choose higher vacation rate i.e. lower vacation time so that the waiting time of customers can be reduced. 6. Conclusions The present investigation on batch arrival queueing system with (i) multiphase service, (ii) multi-phase repair and (iii) multi-phase vacation explores more versatile congestion scenarios due to choice of optional phases apart from essential phase; such situations may also be realized at many places including doctor’s clinic, message transmission, etc.. With the incorporation of N- policy, our study will be helpful in tackling realistic queueing situations encountered in computers and communication networks, manufacturing and production systems, information processing systems, etc. in a cost effective manner. Such type of congestion situations can be seen at various malls wherein the customers arrive in batches and may opt for number of optional services like seeing movie
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or eating some delicious food of their choice, etc. along with essential service like buying household goods and other items of daily need. The server may be unreliable and has a choice either to go for a vacation or to continue the service of the next customer. Other variations of our model are possible by incorporating the concepts of priority, bulk service, retrial, etc. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
W. J. Gray et al, Appl. Math. Model. 24, 391-400, 2000. K. H. Wang et al, Appl. Math. Comput. 165, 45-61, 2005. M. Jain and P. K. Agrawal, Qual. Tech. Quant. Manage. 4, 455-470, 2007. J.C. Ke and C.H. Lin, J. Comput. Appl. Math. 221, 1-15, 2008. G. Choudhury et al, J. Comput. Appl. Math. 231, 349-364, 2009. J. C. Ke and F.M. Chang, Appl. Math. Model. 33, 3186-3196, 2009. G. Choudhury and K.C. Madan, Math. and Comput. Model. 42, 71-85, 2005. G. Choudhury and M. Paul, Stochast. Anal. and Appl. 24, 1-21, 2006. G. Choudhury et al, Appl. Math. Model. 31, 1079-1091, 2007. G. Choudhury, Appl. Math. Comput. 188, 1455-1466, 2007. J. Wang and J. Li, Qual. Tech. Quant. Manage. 5, 179-192, 2008. S. Kumar and Arumuganathan, Qual. Tech. Quant. Manage. 5, 145-160, 2008. K. C. Madan, Queueing Systems 34, 37-46, 2000. J. Medhi, Queueing Systems 42, 239-242, 2002. G. Choudhury, Top 11, 141-150, 2003. G. Choudhury and P. Madhuchanda, Rev. Invest. Oper. 25, 217-228, 2004. G. Choudhury and M. Paul, Stochast. Anal. and Appl. 24, 1-21, 2006. Madan and Al-Rawwash, Appl. Math. Comput. 160, 909-919, 2005. J. C. Ke, Appl. Math. Model. 31, 236-244, 2007. M. Jain and S. Upadhyaya, Appl. Math. Comput. 165, 45-61, 2010.
