A Single Server Queue with Two Phases of ...

Information and Management Sciences Volume 16, Number 2, pp.1-16, 2005

A Single Server Queue with Two Phases of Heterogeneous Service under Bernoulli Schedule and a General Vacation Time Kailash C. Madan

Gautam Choudhury

AHLIA University

Institute of Advanced Study

Kingdom of Bahrain

in Science and Technology India

Abstract We consider a single server queue with Poisson input, two phases of heterogeneous service with Bernoulli schedule and a general vacation time, where the server provides two phases of heterogeneous service one after the other to the arriving customers. After completion of both phases of service the server either goes for a vacation with probability θ(0 ≤ θ ≤ 1) or may continue to serve the next unit, if any, with probability (1 − θ). Otherwise, it remains in the system until a customer arrives. For this model, we first obtain the steady state probability generating functions for the queue size distributions at a random epoch as well as at a departure epoch. Next, we derive the Laplace Stieltjes transform of the waiting time distribution. Finally, we obtain some system performance measures and discuss some important particular cases of this model.

Keywords: M/G/1 Queue, Two Phases of Heterogeneous Service, Bernoulli Schedule, Generalized Vacation Time, Queue Size, Waiting Time and Busy Period. 1. Introduction The single server queueing system with Bernoulli vacation is not new. Keilson and Servi [8] were first to study such a model, where after each service completion the server takes a vacation with probability θ and starts a new service with probability (1 − θ). Subsequently, Keilson and Servi [9], Ramaswamy and Servi [14], Doshi [4, 5] and Takagi [15] among others have studied this and the models of similar nature due to its numerous applications in many real life situations. Recently Madan [11, 12] has studied two similar types of vacation models for the M/G/1 queueing system. In both the models, he introduced the concept of two stage Received May 2004; Revised June 2004 and Accepted December 2004. Supported by the Department of Atomic Energy, Govt. of India, NBHM Project No. 48/2/2001/ R&DII/1211.

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heterogeneous service and Bernoulli schedule along with a single vacation policy. By applying supplementary variable technique he obtained the probability generating functions (PGF) of the queue size distributions at different stages of service as well as for vacation period for both the models. In addition, he cited some important applications to transportation systems and production systems. In fact, Choudhury [1] has also considered the vacation model under a single vacation policy recently. Further, in a most recent paper, Choudhury [2] studied a queueing system with two different vacation periods. However, a two phase queueing system with general service times have been considered by Doshi [6]. The motivation for this type of model comes from some computer and communication networks where messages are processed in two phases by a single server. However, in this paper we propose to study such a model where, after two successive phases of service the server may take a vacation of random length or may decide to stay in the system to provide service to the next customer, if any. The examples of this kind of situation may well be found in some transportation system in which a ferry driver or a locomotive driver may like to go on a vacation after every round trip. Such a trip essentially involves two phases of service that is trip to a particular destination and back to starting point. Next consider another situation of a production system, where the machine producing certain items may require two phases of service in succession such as periodic checking (first phase of service) followed by a usual processing (second phase of service) to complete the processing of raw materials. It may so happen that the machines that need to be stopped for overhauling after these two phases of service. This overhauling may be utilized as a vacation time. Although some aspect of two stage heterogeneous service system with Bernoulli schedule vacation have been discussed by Madan [11, 12], some questions still need to be addressed. Thus the purpose of the present paper is to generalize the result obtained in Madan [11, 12] by deriving the steady state queue size distributions at a random as well as at a departure epoch for generalized vacation time. In addition, we derive the Laplace Stieltjes Transform (LST) of the waiting time distribution. Finally, we obtain some important performance measures and some particular cases of this model, which may lead to remarkable simplification while solving other similar types of queueing models.

A Single Server Queue with Two Phases of Heterogeneous Service

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2. The System We consider an M/G/1 queueing system, where arrivals occur according to a Poisson process with arrival rate ‘λ’ and the server provides two phases of heterogeneous service in succession. The service discipline is assumed to be first come, first served (FCFS). Each customer is first provided the first phase of service (FPS) after completion of which the server immediately starts the second phase of service (SPS) for the same customer. Assuming that the service time B 1 , B2 for two phases are independent random variables follows general law of distribution with distribution function (DF) LaplaceStieltjes transform (LST) Bi∗ (s) and finite moments E(Bik ), k ≥ 1, i = 1, 2. As soon as the SPS of a unit is completed, the server may go for a vacation of random length V with probability θ(0 ≤ θ ≤ 1) or it may continue to serve the next unit, if any, with probability (1 − θ), otherwise, it remains in the system and waits for a new arrival. We further assume that the vacation time random variable V has a general law of distribution with DF V (x), LST V ∗ (s) and finite moments E(V k ) k ≥ 1 independent of the service time random variables. Therefore, the time required by an unit to complete the service cycle, which may be called as modified service time is given by  B + B + V 1 2 B= B + B 1

2

with probabilityθ with probability(1 − θ)

Thus the LST B ∗ (s) of ‘B’ is given by

B ∗ (s) = (1 − θ)B1∗ (s)B2∗ (s) + θB1∗ (s)B2∗ (s)V ∗ (s) with the mean E(B) = E(B1 ) + E(B2 ) + θE(V ) so that λE(B) = λ[E(B1 ) + E(B2 ) + θE(V )] is the utilization factor is denoted by ‘ρ’. Now, since statistical equilibrium condition for queues with vacation and without vacation are the same (e.g. see Doshi [4]), we further assume that ρ < 1 throughout the paper. Further, our model is denoted by M/(G 1 , G2 )/V /1 (BS) queue, where V represents vacation time and BS represents Bernoulli schedule.

3. Queue Size Distribution at Stationary Point of Time In this section, we first set up the system state equations for its stationary queue size (excluding the one being served, if any) distribution by treating the elapsed FPS time,

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the elapsed SPS time and the elapsed vacation time as supplementary variables. Then we solve the equations and derive the probability generating function (PGF) of it. We now define B10 (t)–the elapsed FPS time at time ‘t’, B20 (t)–the elapsed SPS time at time ‘t’, V 0 (t)–the elapsed vacation time at time ‘t’, NQ (t)–the queue size at time ‘t’, Now let us introduce the following random variables

Y (t) =

   0;      1;   2;     

3;

if the server is idle at time ‘t’ if the server is busy with FPS at time ‘t’ if the server is busy with SPS at time ‘t’ if the server is on vacation at time ‘t’

Thus the supplementary variables B 10 (t), B20 (t) and V 0 (t) are introduced in order to obtain a bivariate Markov process {N Q (t), L(t)}, where L(t) = 0 if Y (t) = 0, L(t) = B 10 (t) if Y (t) = 1, L(t) = B20 (t) if Y (t) = 2, and L(t) = V 0 (t) if Y (t) = 3. Next, we define the following probabilities Qn (x; t)dx = Prob[NQ (t) = n, L(t) = V 0 (t); x < V 0 (t) ≤ x + dx],

x > 0, n ≥ 0,

Pi,n (x; t)dx = Prob[NQ (t) = n, L(t) = Bi0 (t); x < Bi0 (t) ≤ x + dx],

x > 0, n ≥ 0, for

i = 1, 2 and R0 (t) = Prob[NQ (t) = 0, L(t) = 0]. Further, it is also assumed that V (0) = 0, V (∞) = 1, B i (0) = 0 and Bi (∞) = 1 for i = 1, 2 and that V (x) and Bi (x) are continuous at x = 0, so that dV (x) , 1 − V (x) dBi (x) , µi (x)dx = 1 − Bi (x) v(x)dx =

i = 1, 2

are the first order differential functions (hazard rate function) of V and B i (i = 1, 2) respectively. Now the analysis of the limiting behaviour of this queueing process at a random epoch can be performed with the help of Kolmogorov forward equations provided R0 = lim R0 (t), t→∞

Pi,n (x)dx = lim Pi,n (x; t)dx t→∞

i = 1, 2, x > 0, n ≥ 0

A Single Server Queue with Two Phases of Heterogeneous Service

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and Qn (x)dx = limt→∞ Qn (x; t)dx, x > 0, n ≥ 0, exist and are independent of the initial state. Then the Kolmogorov forward equations, to govern the system under the steady state conditions (e.g. see Cox [3]) can be written as follows: d P1,n (x) + [λ + µ1 (x)]P1,n (x) = λP1,n−1 (x); n ≥ 0 x > 0, dx d P2,n (x) + [λ + µ2 (x)]P2,n (x) = λP2,n−1 (x); n ≥ 0 x > 0, dx d Qn (x) + [λ + v(x)]Qn (x) = λQn−1 (x); n ≥ 0 x > 0, dx Z Z ∞

λR0 =

(3.2) (3.3)

∞

v(x)Q0 (x)dx + (1 − θ)

0

(3.1)

µ2 (x)P2,0 (x)dx,

0

(3.4)

where P1,−1 (x) = 0, P2,−1 (x) = 0 and Q−1 (x) = 0 in the above equations (3.1), (3.2) and (3.3) respectively. These set of equations are to be solved under the following boundary conditions at x = 0: P1,0 (0) = λR0 + (1 − θ) P1,n (0) = (1 − θ) P2,n (0) =

Z

Qn (0) = θ

Z

Z

∞ 0

µ2 (x)P2,1 (x)dx +

∞ 0

µ2 (x)P2,n+1 (x)dx +

Z

Z

∞ 0

v(x)Q1 (x)dx,

(3.5)

∞ 0

v(x)Qn+1 (x)dx,

n ≥ 1,

(3.6)

∞

µ1 (x)P1,n (x)dx,

0

Z

n ≥ 0,

(3.7)

∞ 0

µ2 (x)P2,n (x)dx,

n ≥ 0,

(3.8)

and the normalizing condition R0 +

2 X ∞ Z X

∞

i=1 n=0 0

Pi,n (x)dx +

∞ X

Qn (x)dx = 1.

n=0

Let us define the following PGF’s: Pi (x; z) = Pi (0; z) = Q(x; z) = Q(0; z) =

∞ X

n=0 ∞ X

n=0 ∞ X

n=0 ∞ X

n=0

z n Pi,n (x);

(: |z| ≤ 1, x > 0 :) for i = 1, 2,

z n Pi,n (0);

(: |z| ≤ 1 :) for i = 1, 2,

z n Qn (x);

(: |z| ≤ 1, x > 0 :),

z n Qn (x);

(: |z| ≤ 1 :).

(3.9)

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Now proceeding in the usual manner with the equations (3.1) — (3.3), we obtain P1 (x; z) = P1 (0; z)[1 − B1 (x)]e−λ(1−z)x ,

x > 0,

(3.10)

P2 (x; z) = P2 (0; z)[1 − B2 (x)]e−λ(1−z)x ,

x > 0,

(3.11)

Q(x; z) = Q(0; z)[1 − V (x)]e−λ(1−z)x ,

x > 0.

(3.12)

Now multiplying equation (3.5) by appropriate powers of z and then taking summation over all values of ‘n’, we get zP1 (0, z) = λR0 (z − 1) + (1 − θ)B2∗ (λ − λz)P2 (0, z) + V ∗ (λ − λz)Q(0, z); R∞

where Bi∗ (λ − λz) = V ∗ (λ − λz) =

−λ(1−z)x dB (x) is the z-transform of B , for i i 0 e R ∞ −λ(1−z)x dV (x) is the z-transform of V , respectively. 0 e

(3.13)

i = 1, 2 and

Proceeding in the similar manner with equations (3.7) and (3.8), we get P2 (0, z) = P1 (0, z)B1∗ (λ − λz),

(3.14)

Q(0, z) = θP2 (0, z)B2∗ (λ − λz).

(3.15)

Now, utilizing equations (3.14) and (3.15) in equation (3.13), we get on simplification λ(z − 1)R0 . [z − ((1 − θ) + θV − λz)B1∗ (λ − λz)B2∗ (λ − λz)]

P1 (0, z) =

∗ (λ

(3.16)

Then from equation (3.10) and (3.16), we get P1 (z) = =

Z

∞

P1 (x, z)dx

0

R0 [1 − B1∗ (λ − λz)] . [((1 − θ) + θV ∗ (λ − λz))B1∗ (λ − λz)B2∗ (λ − λz) − z]

(3.17)

Similarly from equations (3.11), (3.14) and (3.16), we get P2 (z) = =

Z

∞ 0

P2 (x, z)dx

R0 [1 − B2∗ (λ − λz)]B1∗ (λ − λz) . [((1 − θ) + θV ∗ (λ − λz))B1∗ (λ − λz)B2∗ (λ − λz) − z]

(3.18)

Finally from equations (3.12), (3.14), (3.15) and (3.16), we have Q(z) = =

Z

∞

Q(x, z)dx 0

θR0 [1 − V ∗ (λ − λz)]B1∗ (λ − λz)B2∗ (λ − λz) . [((1 − θ) + θV ∗ (λ − λz))B1∗ (λ − λz)B2∗ (λ − λz) − z]

(3.19)

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The unknown constant R0 can be determined by using the normalizing condition (3.9), which is equivalent to R0 + P1 (1) + P2 (1) + Q(1) = 1. Thus we get R0 = (1 − ρ),

(3.20)

which is the steady state probability that the server is idle but available in the system. Also, from equation (3.20) we have ρ < 1, which is the stability condition under which the steady state solution exists. The system state probabilities for this M/(G 1 , G2 )/V /1 (BS) queue can be obtained from equations (3.17), (3.18), (3.19) and (3.20). Thus, we have Prob [the server is busy with FPS]=P 1 (1) = λE(B1 ), Prob [the server is busy with SPS]=P 2 (1) = λE(B2 ), and Prob [the server is on vacation]=Q(1) = λθE(V ) respectively. Let P (z) = P1 (z) + P2 (z) + Q(z) be the PGF of the queue size distribution at a random epoch, then P (z) =

(1 − ρ)[1 − {(1 − θ) + θV ∗ (λ − λz)}]B1∗ (λ − λz)B2∗ (λ − λz) . [((1 − θ) + θV ∗ (λ − λz))B1∗ (λ − λz)B2∗ (λ − λz) − z]

Now let us denote PQ (z) as the PGF of the queue size distribution at departure epoch of this M/(G1 , G2 )/V /1 (BS) queue. Then (e.g. see Kashyap and Chaudhry [7], p.59) we get PQ (z) = R0 + zP (z) (1 − ρ)(1 − z)[(1 − θ) + θV ∗ (λ − λz)]B1∗ (λ − λz)B2∗ (λ − λz) = . [((1 − θ) + θV ∗ (λ − λz))B1∗ (λ − λz)B2∗ (λ − λz) − z]

(3.21)

Taking limit as θ → 0 (i.e. there is no vacation in the system) in the above equation (3.21), we get lim PQ (z) =

θ→0

(1 − ρ)(1 − z)B1∗ (λ − λz)B2∗ (λ − λz) , [B1∗ (λ − λz)B2∗ (λ − λz) − z]

(3.22)

where B = B1 + B2 , ρ = λ[E(B1 ) + E(B2 )] and B ∗ (θ) = B1∗ (s)B2∗ (s). Note that the equation (3.22) is the well known Pollaczek–Khinchine formula for the M/(G1 , G2 )/Vs /1 (BS) queueing model. In fact, Madan [11] has also studied some aspects of this model (for instance see section –6 in his paper). Let LQ be the mean number of customers in the system (i.e. mean queue size) of this M/(G1 , G2 )/Vs /1 (BS) queue, then we have dPQ (z) LQ = dz z=1

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=

λ2 [E(B12 ) + E(B22 ) + 2E(B1 )E(B2 )] + λ[E(B1 ) + E(B2 ) + θE(V )] 2(1 − ρ) θλ2 [E(V 2 ) + 2E(V )(E(B1 ) + E(B2 ))] + (3.23) 2(1 − ρ)

Now suppose that E(B2 ) = 0 and θ = 0 in the above equation (3.23), then we have LQ = ρ +

λ2 E(B12 ) ; 2(1 − ρ)

which is the expression for the mean number of customers in the system of the classical M/G/1 queue (e.g. see Takagi [15], p-7). 4. Waiting Time Distribution In this section, we derive the LST of the waiting time distribution of a test customer for this M/(G1 , G2 )/Vs /1 (BS) queue. To obtain it we follow the simple approach used in Kleinrock [10] for solving for the LST of the time in the system from the PGF of the DF of the number in the system in a regular M/G/1 queue. Let W Q∗ (s) be the LST of the DF of the waiting time of a test customer of this model, then utilizing the standard argument of Kleinrock [10] (for instance see p-197), we may write WQ∗ (λ − λz)B ∗ (λ − λz) = PQ (z)

(4.1)

Now setting s = λ(1 − z) in the above equation (4.1) and utilizing (3.21), we get WQ∗ (s) =

s(1 − ρ) . [s + λ{{(1 − θ) + θV ∗ (s)}B1∗ (s)B2∗ (s) − 1}]

Now having the LST of the waiting time distribution, we can easily derive the LST of the d.f for the response time. The response time ‘W R ’ is the time interval from the arrival time of a tagged customer to the time when it leaves the system after service completion i.e. waiting time plus service time. Now if we denote W R∗ s) as the LST of the DF of ‘WR ’, then WR∗ (s) = WQ∗ (s)B ∗ (s) s(1 − ρ)[θV ∗ (s) + (1 − θ)]B1∗ (s)B2∗ (s) = . [s − λ{1 − {(1 − θ) + θV ∗ (s)}B1∗ (s)B2∗ (s)}] Let E(WR ) be the mean response time of a test customer in this model, then dWR (s) E(WR ) = − ds s=0

A Single Server Queue with Two Phases of Heterogeneous Service

=

9

λ[E(B12 ) + E(B22 ) + 2E(B1 )E(B2 ) + θ[E(V 2 ) + 2E(V ){E(B1 ) + E(B2 )]] 2(1 − ρ) +[E(B1 ) + E(B2 ) + θE(V )]. (4.2)

Note that for E(B2 ) = 0 and θ = 0, we get the mean response time of the regular M/G/1 queueing model. Now if we compare (3.23) with (4.2) then following relationship holds good LQ = λE(WR ). This shows that Little’s formula holds good even for this M/(G 1 , G2 )/V /1 (BS) queue. 5. Expected Busy Period An interesting result, which falls outside the realm of the preceding results, is the expected busy period. Thus in this section an attempt has been made to obtain the expected busy period. To obtain it we follow the argument of alternation renewal process, which seems to be simpler and elegant. We now define the busy period as the length of the time interval that keeps the server busy with FPS and SPS continuously and this continues to the instant when the server becomes free again. This busy period is equivalent to the ordinary busy period generated by the units of the vacation period plus an idle period, which we may call as a generalized idle period. We now define the following notations Tb = length of the busy period and T0 = length of the generalized idle period.] Now Tb and T0 generates an alternating renewal process and therefore we may write E(Tb ) Prob[Tb ] = E(T0 ) 1 − Prob[Tb ]

(5.1)

Clearly, Prob[Tb ] = Prob[The server is busy with FPS] +Prob[The server is busy with SPS] = λ[E(B 1 ) + E(B2 )] 1 and E(T0 ) = θE(V ) + . λ Hence from equations (5.1), (5.2) and (5.3) we have E(Tb ) =

[E(B1 ) + E(B2 )] λθE(V )[E(B1 ) + E(B2 )] + . [1 − λ{E(B1 ) + E(B2 )}] [1 − λ{E(B1 ) + E(B2 )}]

(5.2) (5.3)

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Note that for E(B2 ) = 0 and θ = 0, we have ρ = λE(B1 ) and therefore, E(Tb ) =

E(B1 ) ; (1 − ρ)

which is the mean busy period of the regular M/G/1 queue.

6. Some Particular Cases In this section by specifying vacation time random variable as well as service times random variables we will discuss some important particular cases of this M/(G 1 , G2 )/V /1 (BS) queue. CASE I: M/(G1 , G2 )/V /1 (BS) queue with Erlangian vacation time. Suppose that vacation time is an k-Erlang i.e. E k with probability density function (kv)k xk−1 e−kvx dV (x) = , dx (k − 1)!

v(x) =

V ∗ (λ − λz) =

Then



k > 0, k ≥ 1.

vk vk + λ(1 − z)

k

(6.1) and E(V ) =

1 v

and ρ = λ[E(B1 ) + E(B2 ) + (θ/v)] (< 1) and therefore from equations

(3.17)–(3.19) and (3.20), we have P1 (z) =  P2 (z) = 

(1 − ρ)[1 − B1∗ (λ − λz)] 

1−θ 1−

h

vk λ(1−z)+kv

B1∗ (λ − λz)B2∗ (λ − λz) − z



(1 − ρ)[1 − B2∗ (λ − λz)]B1∗ (λ − λz) 

1−θ 1−

h

vk λ(1−z)+kv



θ(1 − ρ) 1 − Q(z) = 

ik 



1−θ 1−

h

h

ik 

vk λ(1−z)+kv

vk λ(1−z)+kv

B1∗ (λ

ik 

ik 

−

λz)B2∗ (λ

− λz) − z



(6.2)

(6.3)

B1∗ (λ − λz)B2∗ (λ − λz)

B1∗ (λ − λz)B2∗ (λ − λz) − z



(6.4)

Further from equation (3.21), we get the PGF of the queue size distribution at the departure epoch as follows 

(1 − ρ)(1 − z) 1 − PQ (z) = 



1−θ 1−

h

h

vk λ(1−z)+kv

vk λ(1−z)+kv

ik 

ik 

B1∗ (λ − λz)B2∗ (λ − λz)

B1∗ (λ − λz)B2∗ (λ − λz) − z



(6.5)

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CASE II: M/(G1 , G2 )/V /1 (BS) queue with Deterministic vacation time. Recently, Madan [11] considered such a model, where he assumed that vacation time random variable is deterministic. Again it is well known to us that as k → ∞, the whole mass of the distribution Ek tends to concentrate at a point d = ( v1 ) i.e. Ek = D, as k → ∞. Then we have V ∗ (s) → e−ds ; So that from equation (6.1) we have V ∗ (λ − λz) → e−d(1−z)λ and ρ = λ[E(B1 ) + E(B2 ) + θd](< 1). Now taking limit k → ∞, in equations (6.2) and (6.3), we have (1 − ρ)[1 − B1∗ (λ − λz)]

, 1 − θ 1 − e−λ(1−z)d B1∗ (λ − λz)B2∗ (λ − λz) − z (1 − ρ)[1 − B2∗ (λ − λz)]B1∗ (λ − λz)

; P2 (z) =  1 − θ 1 − e−λ(1−z)d B1∗ (λ − λz)B2∗ (λ − λz) − z

P1 (z) = 

(6.6) (6.7)

which agrees with equations (42) and (43) of Madan [11]. Similarly from equation (6.4), we have Q(z) = 

θ(1 − ρ)[1 − e−λ(1−z)d ]B1∗ (λ − λz)B2∗ (λ − λz)

; 1 − θ 1 − e−λ(1−z)d B1∗ (λ − λz)B2∗ (λ − λz) − z

Q(z) = 

θ(1 − ρ)B1∗ (λ − λz)B2∗ (λ − λz)

 1 − θ 1 − e−λ(1−z)d B1∗ (λ − λz)B2∗ (λ − λz) − z

(6.8)

whereas Madan [11] obtained this result [see equation (44)] in our notation as (6.9)

which does not agree with equation (6.8) of our result.

Again, if we consider the PGF of the queue size distribution at a departure epoch of this model then from equation (6.8), we have (1 − ρ)(1 − z)[1 − θe−λ(1−z)d ]B1∗ (λ − λz)B2∗ (λ − λz)

 PQ (z) =  1 − θ 1 − e−λ(1−z)d B1∗ (λ − λz)B2∗ (λ − λz) − z

(6.10)

Further, if we consider M/G/1 queue under Bernoulli vacation schedule without the second phase of service, then with B 2∗ (λ − λz) = 1 and ρ = λ[E(B1 ) + θd] , equation (6.10) yields. (1 − ρ)(1 − z)[1 − θe−λ(1−z)d ]B1∗ (λ − λz)

 PQ (z) =  1 − θ 1 − e−λ(1−z)d B1∗ (λ − λz) − z

(6.11)

which agrees with the result obtained by Madan [13]. Thus the equation (44) of Madan [11] is not a correct expression for Q(z). Therefore, the appropriate expression should be (6.8) in our result.

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CASE III: M/(G1 , G2 )/V /1 (BS) queue with Exponential vacation time. As special case of this model, we now consider the case of Markovian vacation time random variable with mean E(V ) = ( v1 ) and ρ = λ[E(B1 ) + E(B2 ) + (θ/v)] (< 1). The PGF of the various queue size distribution of this model can be obtained by putting k = 1 in equations (6.2) – (6.4). Now by putting k = 1 in above equations (6.2), (6.3) and (6.4), we get (1 − ρ)[v + λ(1 − z)][1 − B1∗ (λ − λz)] [[v + λ(1 − z)(1 − θ)]B1∗ (λ − λz)B2∗ (λ − λz) − z[v + λ(1 − z)]] (1 − ρ)[v + λ(1 − z)][1 − B2∗ (λ − λz)]B1∗ (λ − λz) P2 (z) = [[v + λ(1 − z)(1 − θ)]B1∗ (λ − λz)B2∗ (λ − λz) − z[v + λ(1 − z)]] (1 − ρ)θλ(1 − z)B1∗ (λ − λz)B2∗ (λ − λz) Q(z) = [[v + λ(1 − z)(1 − θ)]B1∗ (λ − λz)B2∗ (λ − λz) − z[v + λ(1 − z)]]

P1 (z) =

(6.12) (6.13) (6.14)

which are the results obtained earlier by Madan [12] for M/(G 1 , G2 )/V /1/(BS) queue having Markovian vacation time. CASE IV: Similarly, we can consider special cases of service times, such as Erlangian, deterministic and exponential. We now summarize the expressions for the PGF of departure point queue size distribution only for different queueing models as given below: 1. For the model M/(Ek1 , Ek2 )/V /1 (BS) queue with generalized vacation time, the PGF of departure point queue size distribution is obtained as PQ (z) =

(µ1 )k1 (µ2 )k2 (1 − ρ)(1 − z)[(1 − θ) + θV ∗ (λ − λz)] , (µ1 )k1 (µ2 )k2 {(1 − θ) + θV ∗ (λ − λz)} − z{µ1 + λ(1 − z)}k1 {µ1 + λ(1 − z)}k2 (6.15)

where the utilization factor ρ = λ[ µ11 +

1 µ2

+ θE(V )].

2. For the model M/(D1 , D2 )/V /1 (BS) queue with generalized vacation time, the PGF of departure point queue size distribution is obtained as PQ (z) =

(1 − ρ)(1 − z)(1 − θ) + θV ∗ (λ − λz)]e−λ(1−z)(d1 +d2 ) . [e−λ(1−z)(d1 +d2 ) {(1 − θ) + θV ∗ (λ − λz)} − z]

where the utilization factor ρ = λ[d 1 + d2 + θE(V )] and di =

1 µi

(6.16)

for i = 1, 2.

3. For the model M/(M1 , M2 )/V /1 (BS) queue with generalized vacation time, the PGF of departure point queue size distribution is obtained as PQ (z) =

(1 − ρ)(1 − z)[(1 − θ) + θV ∗ (λ − λz)]µ1 µ2 . µ1 µ2 {(1 − θ) + θV ∗ (λ − λz)} − z{µ1 + λ(1 − z)}{µ1 + λ(1 − z)}

(6.17)

A Single Server Queue with Two Phases of Heterogeneous Service

where the utilization factor ρ = λ[ µ11 +

1 µ2

13

+ θE(V )].

7. A Numerical Example In order to see the effect of parameter ‘θ 0 ’ on mean queue size LQ of our model here we compute some numerical results along with some graphs for different arrival rates for the mean queue size. For the sake of convenience, we assume that the FPS and SPS time random variables follow exponential distributions with means E(B 1 ) = 1/8, E(B2 ) = 1/16. With these arbitrary values, we compute the values of L Q for two different values of λ and also two different values of vacation time random variable which has also been assumed to have an exponential distribution with mean E(V ) = 1/20 and E(V ) = 1/30 in two different tables respectively. Now based on our results obtained in equations (3.23), we present some numerical results for L Q for λ = 2 and 4 and θ 0 = 0(.1)1 in the following Table 1 and Table 2 and for same values of E(B 1 ), E(B2 ) and E(V ) we plot two graphs for different values of λ and θ. Table 1. Values of LQ for λ = 2

θ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Values of LQ For E(V ) For E(V ) = 1/20 = 1/30 0.54999 0.54999 0.56975 0.56331 0.58983 0.57676 0.61024 0.59037 0.63102 0.60413 0.65217 0.61806 0.67371 0.63214 0.69567 0.64640 0.71806 0.66084 0.54093 0.67546 0.76428 0.69027

Table 2. Values of LQ for λ = 4

θ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Values of LQ For E(V ) For E(V ) = 1/20 = 1/30 2.49992 2.49992 2.74600 2.66158 3.03514 2.84095 3.38095 3.04141 3.80341 3.26723 4.33320 3.52395 5.01985 3.81885 5.94891 4.16167 7.28200 4.56578 9.36543 5.05003 13.09961 5.64189

The above tables clearly show that as θ or λ increases the mean queue size increases. Further it is observed that rate of increase in L Q for E(V ) = 1/20 is faster than that for E(V ) = 1/30. This implies that the expected number of customers in the system is less when the mean duration of vacation is less. Also, we observe that for higher values of λ, the rate of increase in LQ is faster than the lower values of for various values of for

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both the cases, as it should be. In fact, our graphical representations clearly reveal these trends. Figure 1. For E(V ) = 1/20.

Figure 2. For E(V ) = 1/30.

A Single Server Queue with Two Phases of Heterogeneous Service

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Acknowledgement The authors are grateful to the managing editor and the two referees for their valuable comments and suggestions to revise the paper in the present form.

References [1] Choudhury, G., A batch arrival queue with a vacation time under single vacation policy, Computers and Operations Research, Vol.29, pp.1941-1955, 2002. [2] Choudhury, G., Some aspects of M/G/1 queue with two different vacation times under multiple vacation policy, Stochastic Analysis and Applications, Vol.20, No.5, pp.901-909, 2002. [3] Cox, D. R., The analysis of non-Markovian stochastic processes by inclusion of supplementary variables, Proceedings of Cambridge Philosophical Society, Vol.51, pp.433-441, 1955. [4] Doshi, B. T., Queueing systems with vacations — A survey, Queueing Systems, Vol.1, pp.29-66, 1986. [5] Doshi, B. T., Single server queues with vacations, Stochastic Analysis of Computer and Communication Systems (Ed. H. Takagi), Elsevier Science, North-Holland, pp.217-265, 1990. [6] Doshi, B. T., Analysis of a two phase queueing system with general service times, Operations Research Letters, Vol.10, pp.265-272, 1991. [7] Kashyap, B. R. K,. and Choudhry, M. L., An Introduction to Queueing Theory, Aarkay, Calcutta, India, 1988. [8] Keilson, J. and Servi, L. D., Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules, Journal of Applied Probability, Vol.23, pp.790-802, 1986. [9] Keilson, J. and Servi, L. D., Dynamics of the M/G/1 vacation model, Operations Research, Vol.35, pp.575-582, 1987. [10] Kleinrock, L., Queueing Systems, Vol.1, Wiley, New York, 1975. [11] Madan, K. C., On a single server queue with two stage heterogeneous service and deterministic server vacations, International Journal of Systems Sciences, Vol.32, pp.837-844, 2001. [12] Madan, K. C., On a single server queue with two stage general heterogeneous service and binomial schedule server vacations, The Egyptian Statistical Journal, Vol.44, pp.39-55, 2000. [13] Madan, K. C., An M/G/1 queue with optional deterministic server vacations time, Meton, LVII, No.3-4, pp.83-95, 1999. [14] Ramaswami, R. and Servi, L. D., The busy period of the M/G/1 vacation model with a Bernoulli schedule, Stochastic Models, Vol.4, pp.507-521, 1988. [15] Takagi, H., Queueing Analysis: A Foundation of Performance Evaluation, Vol.1, Elsevier Sciences, North-Holland, Amsterdam, 1991.

Authors’ Information Kailash C. Madan holds his M.Sc. in Mathematics and Ph.D in Statistics from Kurukshetra University, Kurukshetra, Haryana, India. Presently, he is a Professor in the college of Mathematical Sciences and Information Technology, Ahila University, Bahrain. His areas of interests are Mathematical Statistics, Applied Stochastic Process, Operations Research and Queueing thoery. He has numerous publications in various International Journals. College of Mathematical Sciences an IT, AHLIA University, P.o.Box.10878, Manama, Kingdom of Bahrain. E-mail: kailashmadan @ hotmail.com

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Gautam Choudhury holds his M.Sc. and Ph.D. in Statistics from Gauhati University, Guwahati, Assam, India. Presently, he is working as an Assistant Professor in Mathematical Sciences Division, Institute of Advanced Study in Science and Technology. His areas of interset are Queueing thoery , Stochastic modelling in Communication Systems and Applied Stochastic Process. He has numerous publications in various journals of Statistics, Mathematics and Operations Research. He is an Associate Editor of “Far East Journal of Theoretical Statistics”. Mathematical Sciences Division , Institute of Advanced Study in Science and Technology, Paschim Boragaon, Guwahati - 781035, Assam, India. E-mail: choudhuryg @ yahoo.com

Tel: + 91 361 2513123