1 unreliable server queue with two phases of service and Bernoulli vacation schedule

Mathematical and Computer Modelling 54 (2011) 673–688

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The optimal control of an M x /G/1 unreliable server queue with two phases of service and Bernoulli vacation schedule Gautam Choudhury a,∗ , Lotfi Tadj b a

Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boragaon, Guwahati – 781035, Assam, India

b

Sobey School of Business, Saint Mary’s University, Halifax, Nova Scotia, B3H 3C3, Canada

article

info

Article history: Received 3 April 2010 Received in revised form 4 March 2011 Accepted 4 March 2011 Keywords: Random breakdown Repair time Delay time N-policy M X /G/1 queue Reliability index

abstract This paper deals with an M X /G/1 Bernoulli vacation queue with two phases of service and unreliable server under N-policy. While the server is working with any phase of service, it may break down at any instant and the service channel becomes unavailable. The breakdown period is followed by a delay period. If no customer arrives when the server is unavailable, the server becomes idle in the system until the queue size builds up to a threshold value N (≥1). As soon as the queue size becomes at least N, the server immediately begins to serve the waiting customers in two successive phases of service. The first phase of service is followed by a second phase, after the completion of which, the server may take a vacation or may remain in the system to serve the next unit, if any. We derive the queue size distribution at a random epoch and at a departure epoch, as well as various system performance measures. Finally, we derive a simple procedure to obtain the optimal stationary policy under a linear cost structure. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction The study of queueing models with service breakdowns or some other kind of interruptions dates back to 1950s. Among some early papers in this area, we refer the readers to the papers by White and Christie [1], Heathcote [2], Keilson [3], Gaver [4], Avi-Itzhak and Naor [5], Thirurengadan [6] and Mitrany and Avi-Itzhak [7] for some fundamental works. Sengupta [8], Ibe and Trivedi [9], Li et al. [10], Tang [11], Takin and Sengupta [12], Madan [13], Li and Lin [14], Fiems et al. [15], Krishnamoorthy et al. [16], and others have studied some queueing systems with interruptions wherein one of the underlying assumptions is that as soon as the service channel fails, it instantaneously undergoes repairs. Further, recently Ke [17–20] investigated some control policies for an unreliable server, i.e., for systems with possible breakdowns. All these papers assume that the server is immediately repaired upon failure. However, in many real-life situations, it may not be feasible to start the repairs immediately due to the non-availability of the server, in which case the system may be turned off. As related works, we should mention the paper of Chao [21] which dealt with queueing systems in which a catastrophe or disaster removes all the work present in the system. A disaster can be viewed as a general breakdown which causes all the jobs in the system to be lost. For example, if a public telephone breaks down, all the customers in the waiting line leave the telephone booth. This type of model has many applications in day-to-day life. Another typical example is the distributed database system with site failure considered by Towsley and Tripathi [22]. The classical vacation scheme with Bernoulli service discipline was initiated and developed significantly by Keilson and Servi [23] and co-workers. Kella [24] suggested a generalized Bernoulli scheme according to which a single server goes

∗

Corresponding author. Tel.: +91 3612520123. E-mail addresses: [email protected] (G. Choudhury), [email protected] (L. Tadj).

0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.03.010

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on k consecutive vacations with probability pk if the queue upon his return is empty. In fact, various aspects of Bernoulli vacation models have been discussed by a number of authors. Recently, there has been considerable attention paid to the study of M /G/1type queueing systems with two phases of service under Bernoulli vacation schedule and different vacation policies, see for instance [25–28], in which after two successive phases of service, the server may go for a Bernoulli vacation. The motivation for these types of models comes from some computer networks and telecommunication systems, where messages are processed in two stages by a single server. As modern telecommunication systems become more complex and the processing power of microprocessors becomes more expensive, the advantages of more sophisticated scheduling becomes more apparent. The need for scheduling the allocation of resources among two or more heterogeneous types of tasks also arises in many other applications. In most of the previous studies, it is assumed that the server is available in the service station on a permanent basis and the service station never fails. However, these assumptions are unrealistic in practice. In practice, we often meet cases where service stations fail and are repaired. Similar phenomena always occur in the area of computer communication networks, flexible manufacturing systems, etc. Because the performance of such systems may be heavily affected by the service station breakdown and delay in repair, these systems with a repairable service station are well worth investigating from the queueing theory viewpoint, as well as from the reliability point of view. Hence, Li et al. [29] considered the reliability analysis of a model under Bernoulli vacation schedule with the assumption that the server is subject to breakdowns and repairs. However, in this present paper, our purpose is to investigate such a type of M /G/1 unreliable server queue with two phases of service and Bernoulli vacation schedule under N-policy, where concepts of repair time and delay time are also introduced. The first study of a batch arrival queue with N-policy was done by Lee and Srinivasan [30]. They presented a procedure to obtain the optimal stationary operating policy under a suitable linear cost structure along with other results. Later, Lee et al. [31] studied this model extensively through different techniques. In fact, some aspects of models of similar nature have also been studied by Lee et al. [32,33], Chae and Lee [34], and Choudhury and Borthakur [35], among others. Further, Choudhury and Madan [26] have investigated such a type of model for a two-stage, batch arrival queue with Bernoulli vacation schedule. Moreover, Tadj et al. [36,37] investigated some bulk service queueing system with Bernoulli vacation schedule under N-policy. The optimal control and management of vacation models has also received considerable attention in the literature, as shown by the survey conducted by Tadj and Choudhury [38]. However, to the best of our knowledge, there is no work that combines batch arrival, two phases of service, Bernoulli vacation, server breakdown, delayed repair, and N-policy in a single queueing model. Thus, in this paper, we propose to study such an M X /G/1 queue with two phases of service and Bernoulli vacation under N-policy for unreliable server, where a breakdown period is followed by a delay period, with a view to unify several classes of related batch arrival queueing systems. To this end, the methodology used will be based on the inclusion of supplementary variables. The remainder of this paper is organized as follows. In Section 2, we give a brief description of the mathematical model. Section 3 deals with the derivation of the stationary distribution of the queue size for the server state at a random epoch. Some important particular cases are provided in Section 4. Derivation of the stationary queue size distribution has been done in Section 5. Section 6 deals with the queue size distribution at a departure epoch. Some important performance measures of this model are derived in Section 7, which may lead to remarkable simplification while solving other similar types of queueing problems. We implement a simple procedure to determine the optimal threshold value ‘N’ under a linear cost structure in Section 8. Finally, we illustrate in the last section the results obtained with some numerical examples. 2. The mathematical model We consider an M x /G/1 queueing system with two phases of heterogeneous service and unreliable server under N-policy, where arrivals occur according to a compound Poisson process with arrival rate λ. The sizes of successive arriving batches are i.i.d. random variables X1 , X2 , . . . , distributed with probability mass function an = Prob{X = n}; n ≥ 1, probability generating function (PGF ) a(z ) = E [Z X ], and the first two factorial moments a[1] and a[2] , respectively. The server remains idle until the queue size reaches a threshold level N (≥ 1). As soon as the queue size exceeds N, the server is turned on to provide each unit two phases of heterogeneous service in succession: the first phase of service (FPS) followed by the second phase of service (SPS) (Busy periods). The service discipline is assumed to be FCFS. The service times random variables Bi , i = 1, 2 are assumed to follow general laws with probability distribution function (d.f.) Bi (x), i = 1, 2, Laplace Stieltjes (k) Transform (LST ) βi∗ (θ ) = E [e−θ Bi ], and finite k-th moments βi , i = 1, 2, where sub-index i = 1 (respectively i = 2) denotes the FPS (respectively SPS). While the server is working with any phase of service, it may break down at any time and the service channel will fail for a short interval of time (Breakdown periods). The breakdowns i.e., server’s life times, are generated by an exogenous Poisson process with rates ε1 for FPS and ε2 for SPS, which we may call some sort of disaster during FPS and SPS periods, respectively. As soon as a breakdown occurs, the server is sent for repair. The customer being served just before the server breaks down waits for the repair to start, which we may refer to as the delay time. The delay time Di of the server for the ith phase of service follows a general law of probability with d.f. Di (y), LST D∗i (θ ) = E [e−θ Di ], and (k)

k-th finite moments di ; for i = 1, 2. The repair times (denoted by Gi for phase i) distributions of the server are assumed to (k)

be arbitrarily distributed with d.f. Gi (y), LST G∗i (θ ) = E [e−θ Gi ], and finite k-th moments gi . Immediately after the server is repaired, it is ready to start its remaining service to customers in either phase of service. The service times are cumulative,

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675

i.e., we consider a preemptive-resume policy for service time, which we may refer to as generalized service times. After each SPS completion, the server may go for a vacation of random length ‘V ’ (vacation period) with probability p(0 ≤ p ≤ 1) or may continue to serve the next unit, if any, with probability q = 1 − p, i.e., the server takes a Bernoulli vacation. The vacation time random variable ‘V ’ follows a general law of probability with d.f. V (y), LST γ ∗ (θ ) = E [e−θ V ], and finite k-th moments γ (k) . The server is turned off as soon as the system becomes empty and it is turned on again when the queue size builds up to level ‘N’. Further, we assume that input process, server’s vacation time, server’s lifetime, server’s repair time, server’s delay time, and service time random variables are mutually independent of each other. The following figure explains the situation.

3. Queue size distribution at a random epoch In this section, we first set up the system state equations for its stationary queue size distribution, by treating elapsed vacation time, elapsed service time, elapsed repair time, and elapsed delay time of the server, for both phases of service, as supplementary variables. Then we solve the equations and derive the PGF s of the stationary queue size distribution. Assume that the system is in steady-state conditions. Let N (t ) be the queue size (including one being served, if any) at time t , V 0 (t ) be the elapsed vacation time of the server, and B0i (t ) be the elapsed service time of the customer for the i-th phase of service at time t, with i = 1, 2 denoting FPS and SPS respectively. In addition, let G0i (t ) and D0i (t ) be the elapsed repair time and elapsed delay time of the server for the i-th phase of service during which breakdown occurs in the system at time t. Further, we introduce the following random variable:

 0,   1,     2,   3, Y (t ) = 4,     5,     6, 7,

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 if the server is waiting for repair during FPS at time t if the server is waiting for repair during SPS at time t if the server is under repair during FPS at time t if the server is under repair during SPS at time t .

The supplementary variables V 0 (t ), B0i (t ), D0i (t ) and R0i (t ) for i = 1, 2 are introduced in order to obtain a bivariate Markov process {N (t ), X (t )}, where X (t ) = 0 if Y (t ) = 0, X (t ) = B01 (t ) if Y (t ) = 1, X (t ) = B02 (t ) if Y (t ) = 2, X (t ) = V 0 (t ) if Y (t ) = 3, X (t ) = D01 (t ) if Y (t ) = 4, X (t ) = D02 (t ) if Y (t ) = 5, X (t ) = G01 (t ) if Y (t ) = 6, and X (t ) = G02 (t ) if Y (t ) = 7. Next we define following limiting probabilities:

ψn = lim Pr {N (t ) = n, X (t ) = 0}; t →∞

n = 0, 1, . . . , N − 1

Hn (y)dy = lim Pr {N (t ) = n, X (t ) = V 0 (t ); y < B0i (t ) ≤ y + dy}; t →∞

x > 0, n ≥ 0

and for i = 1, 2 and n ≥ 1 Pi,n (x)dx = lim Pr {N (t ) = n, X (t ) = B0i (t ); x < B0i (t ) ≤ x + dx}; t →∞

Qi,n (x, y)dy = lim Pr {N (t ) = n, X (t ) =

( x, y ) > 0

Ri,n (x, y)dy = lim Pr {N (t ) = n, X (t ) = G0i (t ); y < G0i (t ) ≤ y + dy|B0i (t ) = x};

(x, y) > 0.

t →∞

(t ); y <

x>0

(t ) ≤ y + dy|B0i (t ) = x};

t →∞

D0i

D0i

Further, it is assumed that V (0) = 0, V (∞) = 1, Bi (0) = 0, Bi (∞) = 1, Di (0) = 0, Di (∞) = 1, Gi (0) = 0, Gi (∞) = 1 for i = 1, 2, and that V (y) is continuous at y = 0 for i = 1, 2; Bi (x) is continuous at x = 0, and Di (y) and Gi (y) are continuous

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at y = 0 for i = 1, 2 respectively, so that dV (y)

α(y)dy = µi (x)dx =

1 − V (y) dBi (x) 1 − Bi (x)

dDi (y)

ηi (y)dy =

;

1 − Di (y)

;

ζi (y)dy =

dGi (y) 1 − Gi (y)

for i = 1, 2

are the first order differential (hazard rate) functions of V , Bi , Di and Gi . 3.1. The steady-state equations The Kolmogorov forward equations to govern the system under steady-state conditions (e.g. see [39]) for i = 1, 2; can be written as follows: d dx d dy d dy d dy

n −

Pi,n (x) + [λ + εi + µi (x)]Pi,n (x) = λ

ak Pi,n−k (x) +

ξi (y)Ri,n (x, y)dy;

n≥1

(3.1)

0

k=1 n −

Hn (y) + [λ + α(y)]Hn (y) = λ(1 − δn,0 )

∞

∫

ak Hn−k (y);

n≥0

(3.2)

ak Qi,n−k (x; y);

n≥1

(3.3)

k=1

Qi,n (x, y) + [λ + ηi (y)]Qi,n (x, y) = λ

n − k=1

Ri,n (x, y) + [λ + ξi (y)]Ri,n (x, y) = λ

n −

ak Ri,n−k (x; y);

n≥1

(3.4)

k=1

λψn = δn,0

∞

∫

α(y)Hn (y)dy + δn,0 q

∞

∫

0

µ2 (x)P2,n+1 (x)dx + λ(1 − δn,0 ) 0

n −

ak ψn−k ;

n = 0, 1, . . . (N − 1), (3.5)

k=1

where δm,n denotes Kronecker’s function and Pi,0 (x) = 0, Qi,0 (x, y) = 0, and Ri,0 (x, y) = 0 for i = 1, 2 occurring in Eqs. (3.1)–(3.4). These sets of equations are to be solved under the boundary conditions at x = 0: P1,n (0) =

∞

∫

α(y)Hn (y)dy + q 0

P1,n (0) =

α(y)Hn (y)dy + q

1≤n≤N −1

∞

∫

0

P2,n (0) =

µ2 (x)P2,n+1 (x)dx;

(3.6)

0

∞

∫

∞

∫

µ2 (x)P2,n+1 (x)dx + λ

N −1 −

0

ψk an−k ;

n≥N

(3.7)

k=0

∞

∫

µ1 (x)P1,n (x)dx;

n≥1

(3.8)

0

and at y = 0: Hn (0) = p

∞

∫

µ2 (x)P2,n+1 (x)dx;

n ≥ 0.

(3.9)

0

Also at y = 0 for i = 1, 2 and fixed values of x Qi,n (x, 0) = εi Pi,n (x);

∫

Ri,n (x, 0) =

x > 0, n ≥ 1

(3.10)

∞

ηi (y)Qi,n (x; y)dy;

x > 0, n ≥ 1.

(3.11)

0

The normalizing condition is N −1 −

ψn +

n =0

∞ ∫ − n=0

∞

∫

∫

+ 0

0

∞

Hn (y)dy + 0

∞

Ri,n (x, y)dxdy

 ∞ − 2 ∫ − n=1 i=1



= 1.

∞

Pi,n (x)dx + 0

∞

∫ 0

∫

∞

Qi,n (x, y)dxdy

0

(3.12)

G. Choudhury, L. Tadj / Mathematical and Computer Modelling 54 (2011) 673–688

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3.2. The model solution To solve the system of Eqs. (3.1)–(3.11), let us introduce the following PGF s for |z | < 1 and i = 1, 2: Qi (x, y; z ) = R i ( x, y ; z ) =

∞ −

z n Qi,n (x; y);

n=1

n =1

∞ −

∞ −

z n Ri,n (x; y);

Ri (x, 0; z ) =

n =1

H (y, z ) =

∞ −

∞ −

z n Qi,n (x; 0)

z n Ri,n (x; 0)

n =1

z n Hn (y);

H (0, z ) =

n =0

P i ( x, z ) =

∞ −

Qi (x, 0; z ) =

∞ −

z n Hn (0)

n =0

z n Pi,n (x);

Pi (0, z ) =

n=1

∞ −

z n Pi,n (0)

n=1

and ψN (z ) = n=0 z ψn Let b(z ) = λ (1 − a(z )). Then, proceeding in the usual manner with Eqs. (3.2)–(3.4), we get a set of differential equations of Lagrangian type whose solutions are given by:

∑ N −1

n

H (y; z ) = H (0; z )[1 − V (y)] exp{−b(z )y};

(3.13)

Qi (x, y; z ) = Qi (x, 0; z )[1 − Di (y)] exp{−b(z )y};

(x, y) > 0 (x, y) > 0 for i = 1, 2 Ri (x, y; z ) = Ri (x, 0; z )[1 − Gi (y)] exp{−b(z )y}; (x, y) > 0 for i = 1, 2

(3.14) (3.15)

where Qi (x, 0; z ) and Ri (x, 0; z ) for i = 1, 2 can be obtained from Eqs. (3.9) and (3.10), which after simplification yield Qi (x, 0; z ) = εi Pi (x; z )

(3.16)

and Ri (x, 0; z ) = Qi (x, 0; z )Di (b(z )). ∗

(3.17)

Now solving the differential equation (3.1), we get Pi (x; z ) = Pi (0; z )[1 − Bi (x)] exp{−ϖi (z )x},

x > 0 for i = 1, 2;

(3.18)

where ϖi (z ) = b(z ) + εi (1 − Gi (b(z ))Di (b(z ))) for i = 1, 2. Utilizing (3.16)–(3.18) in (3.14) and (3.15) respectively, we get for i = 1, 2 ∗

∗

Qi (x, y : z ) = εi Pi (0; z )[1 − Bi (x)] exp{−ϖi (z )x} × [1 − Di (y)] exp{−b(z )y}

(3.19)

Ri (x, y : z ) = εi D∗i (b(z ))Pi (0; z )[1 − Bi (x)] exp{−ϖi (z )x} × [1 − Gi (y)] exp{−b(z )y}.

(3.20)

n

Multiplying Eqs. (3.6) and (3.7) by z and then taking summation over all possible values of n ≥ 1 and utilizing (3.5) by ∑∞ ∑N −1 noting that n=N z n k=0 ψk an−k = ψ0 − λ−1 ψN (z )b(z ), we get on simplification P1 (0, z ) = z −1 qP1 (0; z )β2∗ (ϖ2 (z )) + H (0; z )γ ∗ (b(z )) − ψN (z )b(z ).

(3.21)

Similarly from Eqs. (3.7) and (3.8), we have P2 (0, z ) = P1 (0; z )β1∗ (ϖ1 (z )).

(3.22)

H (0, z ) = z

(3.23)

−1

pP2 (0; z )β2 (ϖ2 (z )). ∗

Utilizing (3.22) and (3.23) in (3.21) and simplifying we get P1 (0, z ) =

z ψN (z )b(z ) . [q + pγ ∗ (b(z ))]β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z

(3.24)

Letting z → 1in (3.20), we obtain by L’Hospital’s rule

λa[1] P1 (0, 1) =

[ N −1 ∑

ψn

n =0

(1 − ρH ) (1)

] , (1)

(3.25) (1)

where ρH = ρ1 {1 + ε1 (g1 + d1 )} + ρ2 {1 + ε2 (g2 i = 1, 2 and ρ3 = λa[1] γ (1) . Thus we have

λa[1] p H (y, 1) =

[N −1 ∑

+ d(21) )} + pρ3 is the utilizing factor of the system, ρi = λa[1] βi(1) for

]

ψn [1 − V (y)]

n=0

(1 − ρH )

(3.26)

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G. Choudhury, L. Tadj / Mathematical and Computer Modelling 54 (2011) 673–688

and for i = 1, 2 we have

λa[1]

[ N −1 ∑

] ψn [1 − Bi (x)]

n =0

Pi (x, 1) =

(3.27)

(1 − ρH ) [ N −1 ] ∑ εi λa[1] ψn [1 − Bi (x)][1 − Di (y)] n =0

Qi (x, y, 1) =

(3.28)

(1 − ρH ) [N −1 ] ∑ εi λa[1] ψn [1 − Bi (x)][1 − Gi (y)] n =0

and Ri (x, y, 1) =

.

(1 − ρH )

(3.29)

Further, let us define

ψn = A0 φn ;

n = 0, 1, . . . , (N − 1),

(3.30)

where φn = Pr {A batch of ‘n’ customers arrive in the system during an idle period} and A0 is a normalizing constant that can be determined by utilizing the normalizing condition (3.12), which finally yields after simplification

(1 − ρH ) A0 = [ ]. N −1

∑

(3.31)

ϕn

n=0

Now from the above expression (3.31), we have ρH < 1, which is the stability condition under which the steady-state solution exists. Thus we summarize the above results in the following Theorem 3.1. Theorem 3.1. Under the stability condition ρH < 1, the joint distribution of the number in the queue and the server’s state has the following partial PGFs

[N −1 ] ∑ n (1 − ρH ) z φn n=0 ψN (z ) = , [N −1 ] ∑ φn

(3.32)

n =0

P1 (x; z ) = P2 (x; z ) = H (y; z ) =

zb(z )ψN (z )[1 − B1 (x)] exp {−(ϖ1 (z ))x}

{q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z

,

zb(z )β1∗ (ϖ1 (z ))ψN (z )[1 − B2 (x)] exp{−(ϖ2 (z ))x}

{q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z

(3.33)

,

(3.34)

pb(z )β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z ))ψN (z )[1 − V (y)] exp{−b(z )y}

(3.35)

{q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z

ε1 zb(z )ψN (z )[1 − B1 (x)] exp{−(ϖ1 (z ))x} × [1 − D1 (y)] exp{−(b(z ))y} {q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z ∗ ε2 zb(z )β1 (ϖ1 (z ))ψN (z )[1 − B2 (x)] exp{−(ϖ2 (z ))x} × [1 − D2 (y)] exp{−(b(z ))y} Q2 (x, y; z ) = ; {q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z ... ε1 zD∗1 (b(z ))b(z )ψN (z )[1 − B1 (x)] exp{−(ϖ1 (z ))x} × [1 − G1 (y)] exp{−(b(z ))y} R1 (x, y; z ) = ; {q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z ... ε2 zD∗2 (b(z ))b(z )β1∗ (ϖ1 (z ))ψN (z )[1 − B2 (x)] exp{−(ϖ2 (z ))x} × [1 − G2 (y)] exp{−(b(z ))y} R2 (x, y; z ) = ; {q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z ... Q1 (x, y; z ) =

where ϖi (z ) = b(z ) + εi (1 − Gi (b(z ))Di (b(z ))) for i = 1, 2 and b(z ) = λ(1 − a(z )). ∗

∗

Next we investigate the queue size distributions due to the state of the server.



(3.36)

(3.37)

(3.38)

(3.39)

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679

Theorem 3.2. Under the stability condition ρH < 1, the marginal PGFs of the server’s state queue size distribution are given by

[N −1 ∑

]

(1 − ρH )b(z ) z φn [1 − β1∗ (ϖ1 (z ))] n=0 P1 ( z ) = [ N −1 ] ∑ ϖ1 (z ) φn [{q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z ] n +1

(3.40)

n =0

[N −1 ] ∑ n +1 (1 − ρH )b(z ) z φn β1∗ (ϖ1 (z ))[1 − β2∗ (ϖ2 (z ))] n=0 P2 ( z ) = [ N −1 ] ∑ ϖ2 (z ) φn [{q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z ]

(3.41)

n =0

p(1 − ρH ) H (z ) =

[N −1 ∑

]

z φn β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z ))[1 − γ ∗ (b(z ))] n

n=0

[N −1 ∑

(3.42)

]

φn [{q +

pγ ∗ (b(z ))}β ∗ (ϖ 1

1

(z ))β2 (ϖ2 (z )) − z ] ∗

n=0

[N −1 ] ∑ n+1 ε1 (1 − ρH )(1 − D∗1 (b(z ))) z φn [1 − β1∗ (ϖ1 (z ))] n=0 Q1 (z ) = [ N −1 ] ∑ ϖ1 (z ) φn [{q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z ]

(3.43)

n =0

[N −1 ∑

]

ε2 (1 − ρH )(1 − D2 (b(z ))) z φn β1∗ (ϖ1 (z ))[1 − β2∗ (ϖ2 (z ))] n=0 Q2 (z ) = [ N −1 ] ∑ ϖ2 (z ) φn [{q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z ] ∗

n+1

(3.44)

n =0

[ N −1 ] ∑ n +1 ε1 (1 − ρH )D∗1 (b(z ))(1 − G∗1 (b(z ))) z φn [1 − β1∗ (ϖ1 (z ))] n =0 R1 ( z ) = [N −1 ] ∑ ϖ1 (z ) φn [{q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z ]

(3.45)

n=0

[N −1 ∑

]

ε2 (1 − ρH )D2 (b(z ))(1 − G2 (b(z ))) z φn β1∗ (ϖ1 (z ))[1 − β2∗ (ϖ2 (z ))] n =0 and R2 (z ) = . [ N −1 ] ∑ ϖ2 (z ) φn [{q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z ] ∗

∗

n +1

(3.46)

n =0

Proof. Integrating (3.33), (3.34) and (3.29) with respect to x and y respectively, then

∞

i = 1, 2 and 0 e−sx (1 − V (y))dy = respect to y, we get for i = 1, 2 Qi (x, z ) =

[1−γ ∗ (s)] ; s

∞ 0

e−sx (1 − Bi (x))dx =

[1−βi∗ (s)] s

for

we get formulae (3.40)–(3.42). Similarly, integrating Eqs. (3.36)–(3.39) with

∞

∫

Qi (x, y; z )dy = εi [b(z )]−1 [1 − D∗i (b(z ))]Pi (0; z )[1 − Bi (x)] exp{−(ϖi (z )x)}

(3.47)

0

and Ri (x, z ) =

∞

∫

Ri (x, y; z )dy = εi [b(z )]−1 D∗i (b(z ))[1 − G∗i (b(z ))]Pi (0; z )[1 − Bi (x)] exp{−ϖ (z )x}.

(3.48)

0

Further integrating (3.47) and (3.48) with respect to x, we get formulae (3.43)–(3.46).



Theorem 3.3. Let Pj be the stationary distribution of the number of customers in the queue at a random epoch, then its ∑∞ j corresponding PGF , i.e. PQ (z ) = j=0 z Pj is given by

[N −1 ] ∑ n (1 − ρH )(1 − z ) z φn [q + pγ ∗ (b(z ))]β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) n=0 PQ (z ) = . [N −1 ] ∑ ∗ ∗ ∗ ϕn [{q + pγ (b(z ))}β1 {ϖ1 (z )}β2 (ϖ2 (z )) − z ] n =0

(3.49)

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Proof. The result follows directly from Theorem 3.2 with the help of PGFsψN (z ), H (z ), Pi (z ), Qi (z ) and Ri (z ), since the ∑2 distribution of the number of customers in the queue has PGF PQ (z ) = ψN (z ) + zH (z ) + i=1 {Pi (z ) + Qi (z ) + Ri (z )}.  4. Particular cases In this section we will discuss very briefly some important particular cases of this type of M X /G/1 model, which are consistent with the existing literature. For example, if we take N = 1 (i.e. there is no threshold in the system) in formula (3.49), then we have PQ ( z ) =

(1 − ρH )(1 − z )[q + pγ ∗ (b(z ))]β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) [q + pγ ∗ (b(z ))]β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z

= ζ (z )

(say);

(4.1)

which is the PGF of the queue size distribution at a random epoch for an M x /G/1 queue with two phases of service and Bernoulli vacation schedule for unreliable server, and this is equivalent to a Pollaczek–Khinchin formula for this type of model. Further, if we consider the case of an M X /G/1 queueing system with two phases of service and random breakdown but (1) (1) without delayed repair, then with ρH = ρ1 (1 + ε1 g1 ) + ρ2 (1 + ε2 g2 ) + pρ3 and D∗i (b(z )) = 1 for i = 1, 2, formula (4.1) yields PQ ( z ) =

(1 − ρH )(1 − z )[q + pγ ∗ (b(z ))]β1∗ (b(z ) + ε1 (1 − G∗1 (b(z ))))β2∗ (b(z ) + ε2 (1 − G∗2 (b(z )))) ; [q + pγ ∗ (b(z ))]β1∗ (b(z ) + ε1 (1 − G∗1 (b(z ))))β2∗ (b(z ) + ε2 (1 − G∗2 (b(z )))) − z

...

(4.2)

which is the PGF of the queue size distribution at a random epoch for an M X /G/1 Bernoulli vacation queue with two phases of service and random breakdowns. Note that for ε1 = ε2 = 0 (i.e. there is no breakdown in the system), the above formula (4.2) is consistent with the result obtained by Choudhury and Madan [25]. Again, if we take p = 0 (i.e. there is no Bernoulli vacation in the system) in the above formula (3.49), then it reduces to

[ N −1 ] ∑ n (1 − ρ1 {1 + ε1 (g1(1) + d(11) )} − ρ2 {1 + ε2 (g2(1) + d(21) )})(1 − z ) z φn β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) n =0 PQ ( z ) = ; [N −1 ] ∑ ∗ ∗ φn [β1 (ϖ1 (z )) β2 (ϖ2 (z )) − z ] n=0

...

(4.3)

which is the PGF of the stationary queue size distribution at a random epoch of an M /G/1 queue with two phases of service for an unreliable server under N-policy, where the server provides to each customer two successive phases of service, i.e. FPS followed by a SPS. Note that for ε1 = 0 (i.e. there is no breakdown in the system during FPS) and β2∗ (ϖ2 (z )) = 1, the formula (4.3) is consistent with the formula (3.1) of Lee et al. [31]. Similarly, if we take z = γ ∗ (b(z )) in (3.49), we get on simplification x

(q − ρ1 {1 + ε1 (g1(1) + d(11) )} − ρ2 {1 + ε2 (g2(1) + d(21) )})(1 − z )(q + pz )β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) PQ ( z ) = [ N −1 ] ∑ φn (q + pz )β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z

[N −1 ∑ n=0

z n φn

] ;

n =0

...

(4.4)

which is the PGF of the stationary queue size distribution at the service completion epoch of an M x /G/1 queueing system with two phases of service and Bernoulli feedback mechanism under N-policy. This type of model was first studied by Takács [40] but without breakdown, i.e. for ε1 = 0, β2∗ (ϖ2 (z )) = 1 and N = 1. Also, similar models have been studied by Madan and Rawwash [41] for a reliable server without N-policy.  5. Analysis of the stationary queue size distribution In this section, an attempt has been made to provide an appropriate interpretation for the PGF of the stationary queue size distribution of this model. Now to provide an appropriate interpretation of ψN (z ), let us introduce the following probability

φn ;

n = 0, 1, . . . , (N − 1),

G. Choudhury, L. Tadj / Mathematical and Computer Modelling 54 (2011) 673–688

681

as the probability that a batch of customers finds ‘n’ customers in the system during an idle period, that satisfies the following recursive equations

φn =

n −

ak φn−k ;

n = 1, 2, . . . , (N − 1) and φ0 = 1.

k=1

Let {ϑn ; n = 0, 1, 2, . . . , (N − 1)} be the conditional probability that there is a batch of ‘n’ customers in the system during an idle period given that the server is idle. Then, conditioning the number of arrivals in (3.30), we get

ϑn =

φn ψn = N −1 ; ∑ ∑ ψn φn

n = 0, 1, 2, . . . , N − 1,

N −1 n =0

∑ N −1

(5.1)

n =0



φn gives the mean number of groups arriving during an idle period. The PGF of {ϑn ; n = 0, 1, 2, . . . , (N − 1)} is thus given by

where

n=0

N −1

ϑN (z ) =

N −1 −

∑ z ϑn = n

n=0

z n ϕn

n =0 N −1

∑

.

(5.2)

ϕn

n =0

Hence, from (3.30) and (5.1), we have ψj = (1 − ρH )ϑj ; j = 0, 1, . . . , N − 1, and therefore

(1 − ρH )−1 ψN (z ) = ϑN (z );

(5.3)

which is the relationship between ψN (z ) and ϑN (z ). Thus, the stochastic decomposition property for this model can be demonstrated easily by writing



z φn  (1 − ρH )(1 − z )[q + pγ ∗ (b(z ))]β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z ))   n=0    −1  N∑  [q + pβ2∗ (b(z ))]β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z φn n

∑

PQ (z ) =



N −1

n=0

= ζ (z )ϑN (z ); where ζ (z ) =

(1−ρH )(1−z )[q+pγ ∗ (b(z ))]β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) [q+pβ2∗ (b(z ))]β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z ))−z

(5.4) is the PGF of the stationary queue size at a random epoch of an M x /G/1

queue with two phases of service and Bernoulli vacation for unreliable server [see Eq. (4.1) of Section 4]. Next, we are interested in the system’s state probabilities, which are given in Corollary 5.1.



Corollary 5.1. If the system is in steady-state conditions, then (i) the probability that the server is idle is PI = 1 − ρH (1) (ii) the probability that the server is busy with FPS is PB1 = λa[1] β1

(1)

(iii) the probability that the server is busy with SPS is PB2 = λa[1] β2 (iv) the probability that the server is on vacation is PV = pλa[1] γ (1) (1) (1) (v) the probability that the server is waiting for repair during FPS is PW1 = λa[1] β1 ε1 d1 (1) (1) (vi) the probability that the server is waiting for repair during SPS is PW2 = λa[1] β2 ε2 d2 (1)

(1)

(1)

(1)

(vii) the probability that the server is under repair during FPS is PR1 = λa[1] β1 ε1 g1

(viii) the probability that the server is under repair during SPS is PR2 = λa[1] β2 ε2 g2 . Proof. Since PVi = limz →1 H (z ), PBi = limz →1 Pi (z ), PWi = limz →1 Qi (z ), PRi = limz →1 Ri (z ) for i = 1, 2, and PI = 1−

∑2

i =1

{PBi + PWi + PRi }, the stated formulae follow by direct calculation.



Remark 5.1. From expression (5.4), we observe that the stationary queue size distribution at a random epoch of an M x /G/1 type of queue with two phases of service and Bernoulli vacation schedule for unreliable server under N-policy can be decomposed into distributions of two independent random variables viz.1. The queue size distribution of an M x /G/1 queue with two phases of service and Bernoulli vacation schedule for unreliable server [represented by first term of the right-hand side of the expression (5.4)] and 2. The additional queue size distribution due to N-policy [represented by the second term of the right-hand side of the expression (5.4)]. This confirms the stochastic decomposition property of Fuhrmann and Cooper [42].



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G. Choudhury, L. Tadj / Mathematical and Computer Modelling 54 (2011) 673–688

6. Queue size distribution at a departure epoch In this section, we derive the PGF of the stationary queue size distribution at a departure epoch for this model, as a classical generalization of Choudhury and Madan’s [26] results. The key result of this section is now stated in Theorem 6.1. Theorem 6.1. Under the steady-state condition, the PGF of the steady-state queue size distribution at a departure epoch of an M x /G/1 unreliable server queue with two phases of service and Bernoulli vacation schedule under N-policy is given by

(1 − ρH )

[N −1 ∑

]

z φn [1 − a(z )][q + pγ ∗ (b(z ))]β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) n

n=0

π (z ) = a[1]

[N −1 ∑

]

φn [{q +

; pγ ∗ (b(z ))}β ∗ (ϖ 1

1

(6.1)

(z ))β2 (ϖ2 (z )) − z ] ∗

n=0

Proof. Following the argument of PASTA (see [43]), we state that a departing customer will see ‘j’ customers in the system just after a departure if and only if there were (j + 1) customers in the system just before the departure. Now denoting {πj ; j ≥ 0} as the probability that there are j units in the system at a departure epoch, then for j ≥ 0,

πj = B0 q

∞

∫

µ2 (x)P2,j+1 (x)dx + B0 0

∞

∫

α(y)Hj (y)dy;

(6.2)

0

where B0 is the normalizing constant. Multiplying both sides of Eq. (6.2) by z j , summing over j ≥ 0, and utilizing Eqs. (3.34) and (3.35), we get on simplification

π (z ) =

B0 ψN (z )b(z )[q + pγ ∗ (b(z ))]β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z ))

[q + pγ ∗ (b(z ))]β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z

.

(6.3)

Utilizing the normalizing condition π (1) = 1, we get B0 = [λa[1] ]−1 .

(6.4)

Hence formula (6.1) follows by inserting (6.4) in (6.3).



It should be noted here that for ε1 = ε2 = 0 (i.e. for reliable server), the above formula (6.1) is consistent with the result obtained by Choudhury and Madan [26]. Next, the relationship between the stationary queue size distributions at a random epoch and at a departure epoch is stated in Corollary 6.1. Corollary 6.1. Under the steady-state condition, the relationship between the PGFs of the queue size distributions at a random epoch and at a departure epoch of an M x /G/1 unreliable server queue with two phases of service and Bernoulli vacation schedule under N-policy is given by

π (z ) =

[1 − a(z )] a[1] (1 − z )

PQ (z ) = Ht (z )ζ (z )ϑN (z );

(6.5)

where Ht (z ) is the PGF of the number of customers placed before an arbitrary test customer (tagged customer) in a batch in which the tagged customer arrives. This number is given as the backward recurrence time in the discrete time renewal process where [1−a(z )] renewal points are generated by the arrival size random variable (e.g., see [44], p. 46), i.e., Ht (z ) = a/ (1)(1−z ) and a[1] = a/ (1).  Remark 6.1. From Corollary 6.1, it is observed that queue size distribution at the departure epoch of an M x /G/1 unreliable server queue with two phases of service and Bernoulli vacation schedule under N-policy is the convolution of the distributions of three independent random variables. The first one is the number of units placed before a tagged customer in a batch in which the tagged customer arrives. This is due to the randomness property of the arriving batch size. The interpretation of the other two random variables is provided in Remark 5.1.  Remark 6.2. Now setting z = 0 in Eq. (6.1), we get

π (0) = Pr {No customer is waiting in the system at the departure epoch} (1 − ρH )φ0 = [ N −1 ] = π 0 ∑ a[1] φn n =0

so that π0 a[1] = ψ0 . This exhibits an interesting phenomenon. It states that a random observer is more likely to find the system empty than a departing customer leaving the system. 

G. Choudhury, L. Tadj / Mathematical and Computer Modelling 54 (2011) 673–688

683

Next, we are interested in the distribution of the number of customers in the system immediately after those time points at which a customer departs or an idle period ends. Theorem 6.2. Let Ω (z ) be the PGF of the number of customers in the system at those time points immediately after which a customer departs or an idle period is terminated in an M x /G/1 queue with two phases of service and Bernoulli vacation schedule for unreliable server under N-policy. Then, under the steady-state condition, we have

 (1 − ρH )



N −1

z φj [{q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − za(z )] j

∑ j =0

Ω (z ) =

 (1 + a[1] − ρH )

N −1

∑

.

 φj [{q +

pγ ∗ (b(z ))}β ∗ (ϖ 1

1

(6.6)

(z ))β2 (ϖ2 (z )) − z ] ∗

j=0

Proof. The result follows directly by utilizing the stochastic decomposition property for an M x /G/1 queue with two phases of service and Bernoulli vacation schedule for unreliable server under N-policy viz- (e.g. see Section 4)

Ω (z ) = Ω0 (z )ϑN (z ),

(6.7)

where Ω0 (z ) is the PGF of the stationary queue size distribution at which a customer departs or at the termination epoch of an idle period for an M x /G/1 type of queue with two phases of service and Bernoulli vacation schedule for unreliable server. This can be obtained easily by replacing the original service time distribution by our generalized service time distribution i.e., β ∗ (θ ) = [q + pγ ∗ (θ)]β1∗ (θ )β2∗ (θ ) in terms of LST in the well-known result of Gross and Harris [45], which is of the form [Section 5.1.9, p. 237]:

Ω0 (z ) =

C0 [K (z ) − za(z )]

[K ( z ) − z ]

;

(6.8)

where K (z ) is the PGF of a batch of customers who arrived during our generalized service time B, and C0 is a constant to be evaluated. Clearly, we have K (z ) = [q + pγ ∗ (b(z ))]β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )). Now utilizing K (z ) in (6.8), we may write

Ω0 (z ) =

C0 [{q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − za(z )]

[{q + pγ ∗ (b(z ))}β1∗ (ϖ1 (z ))β2∗ (ϖ2 (z )) − z ]

.

(6.9)

Since Ω0 (1) = 1, we get C0 =

(1 − ρH ) . (1 + a[1] − ρH )

(6.10)

Hence formula (6.6) follows by inserting (6.9), (6.10) and (5.2) in (6.7).



7. System’s performance measures Our next objective is to provide explicit expressions for some important performance measures of the system. In this section, we will first discuss two main reliability indices of the system viz.- the system availability and the failure frequency under the steady-state condition. Suppose that the system is initially empty. Let Av (t ) be the point-wise availability of the server at time ‘t ′ , that is, the probability that the server is either serving a customer or the server is available if the server is free and up during an idle period, such that the steady-state availability of the server will be Av = limt →∞ Av (t ). Theorem 7.1. The steady-state availability of the server and the server’s failure frequency are respectively given by (1)

(1)

(1)

(1)

Av = 1 − ε1 ρ1 (d1 + g1 ) − ε2 ρ2 (d2 + g2 ) − pρ3

(7.1)

and Wf = ε1 ρ1 + ε2 ρ2 .

(7.2)

Proof. The result follows directly by considering the following equations Av =

N −1 − n =0

Since

∞ 0

ψn +

2 ∫ − i −1

[1 − Bi (x)]dx =

∞

Pi (x, 1)dx and 0

∞ 0

Wf = α1

∞

∫

P1 (x, 1)dx + α2 0

(1)

∞

∫

P2 (x, 1)dx. 0

xdBi (x) = βi , we get the required result from (3.27).



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G. Choudhury, L. Tadj / Mathematical and Computer Modelling 54 (2011) 673–688

Next, we are interested in the mean queue size at a random and at a departure epoch. Theorem 7.2. Let E (NQ ) and E (ND ) be the expected number of customers in the queue at a random epoch and at a departure epoch, respectively, then we have

[λa[1] ]2 ⌊β1(2) {1 + ε1 (d(11) + g1(1) )}2 + β2(2) {1 + ε2 (γ2(1) + g2(1) )}2 + pγ (2) ⌋ 2(1 − ρH ) [λa[1] ]2 [ε1 β1(1) {d(12) + g1(2) + 2d(11) g1(1) } + ε2 β2(1) {d(22) + g2(2) + 2d(21) g2(1) }] + 2(1 − ρH ) [λa[1] ]2 [pγ (1) {ρ1 {1 + ε1 (d(11) + g1(1) )} + ρ2 {1 + ε2 (d(21) + g2(1) )}}] ρH a[2] + + (1 − ρH ) 2a[1] (1 − ρH ) N∑ −1       2  (1) (1) (1) (1) jφj 1 + ε2 (d2 + g2 ) λa[1] ρ1 ρ2 1 + ε1 d1 + g1 j =1  + + N∑ −1 (1 − ρH ) φj

E (NQ ) = ρH +

(7.3)

j=0

a[2]

and E (ND ) = E (NQ ) +

2a[1]

.

(7.4)

Proof. The results follow directly by differentiating (3.49) and (6.1) with respect to z, then taking the limit z → 1, and by using the L’Hospital’s rule.  8. Optimal cost structure The determination of an optimal policy for a queueing system is an important issue as shown by the recent survey of Tadj and Choudhury [38]. This is usually done by developing the total expected cost function per unit time for the system and then deriving the relevant optimal system parameters. Thus, in this section, we will determine the optimal value of ‘N’ that minimizes the long run average cost of an M x /G/1 queue with two phases of service with Bernoulli vacation schedule for unreliable server under N-policy. To determine the optimal value of N, we consider the following linear cost structure, similar to that considered in [30] (also see [31]. Let CA (N ) be the average cost per unit of time, then (1)

CA (N ) = {ρ1 (1 + ε1 (g1

+ d(11) )) + ρ2 (1 + ε2 (g2(1) + d(21) )) + pρ3 }Cw + Ch E (NQ ) +

Cs E (Tc )

;

(8.1)

where Cs is the startup cost per cycle, Ch is the holding cost per unit of time, Cw is the operating cost per unit of time, and Tc is the cycle length.   To obtain E (Tc ) let us define T0 as the length of the idle period, and Tb length of the busy period. Since

∑ N −1 j=0

φj gives

the mean number of batches arriving during an idle period with λ as the arrival rate, the mean duration of that number of ∑N −1 ∑N −1 batches is j=0 φj /λ, which gives the mean idle period as E (T0 ) = j=0 φj /λ. Further, the mean number of arrivals during ρ E (T )

an idle period is E (T0 )λa[1] , so that the mean busy period equals E (Tb ) = (1H−ρ 0) . Since E (Tc ) = E (T0 ) + E (Tb ), we have H N −1

∑ E (Tc ) =

φj

j =0

λ(1 − ρH )

.

Inserting (7.4) and (8.2) in (8.1) we get the average cost per unit time as

 CA (N ) = Ch

+

[λa[1] ]2 {β1(2) (1 + ε1 (g1(1) + d(11) ))2 + β2(2) (1 + ε2 (g2(1) + d(21) ))2 + pγ (2) } 2(1 − ρH )

[λa[1] ]2 [ε1 β1(1) {d(12) + g1(2) + 2d(11) g1(1) } + ε2 β2(1) {d(22) + g2(2) + 2d(21) g2(1) }] 2(1 − ρH )

[λa[1] ]2 [ρ1 ρ2 {1 + ε1 (d(11) + g1(1) )}{1 + ε2 (d(21) + g2(1) )}] ρH a[2] + (1 − ρH ) 2a[1] (1 − ρH )  (1) (1) (1) 2 (1) [λa[1] ] [pγ {ρ1 {1 + ε1 (d1 + g1 )} + ρ2 {1 + ε2 (d2 + g2(1) )}}] + (1 − ρH )

+

(8.2)

G. Choudhury, L. Tadj / Mathematical and Computer Modelling 54 (2011) 673–688

685

+ (Ch + Cw )[λa[1] {β1(1) (1 + ε1 (g1(1) + d(11) )) + β2(1) (1 + ε2 (g2(1) + d(21) )) + pγ (1) }] λCs ⌊1 − λa[1] {β1(1) (1 + ε1 (g1(1) + d(11) )) + β2(1) (1 + ε2 (g2(1) + d(21) )) + pγ (1) }⌋ + Ch M (N − 1) ; L(N − 1) ∑k ∑k where L(k) = j=0 ϕj and M (k) = j=1 jϕj . +

(8.3)

In order to determine the optimal value of ‘N’, let us consider the following differences

1CA (k) = CA (k + 1) − CA (k) =

ϕk I (k) L(k − 1)L(k)

;

where ‘∆’ is the difference operator and (1)

(1)

I (k) = Ch [kL(k) − M (k)] − λCs [1 − λa[1] {β1 (1 + ε1 (g1

+ d(11) )) + β2(1) (1 + ε2 (g2(1) + d(21) )) + pγ (1) }];

....

(8.4)

Since [kL(k) − M (k)] > 0 and



φk

L(k−1)L(k)



> 0, then we have 1CA (k + 1) > 0. Let ‘m’ be the first ‘k’ such that I (k) > 0, then

we have I (m + 1) = Ch [(m + 1)L(m + 1) − M (m + 1)]

− λCs [1 − λa[1] {β1(1) (1 + ε1 (g1(1) + d(11) )) + β2(1) (1 + ε2 (g2(1) + d(21) ) + pγ (1) )}] = I (m) + Ch L(m). Thus, we observe that I (m + 1) > I (m). Hence, for some n > m, we have CA (n) > CA (m). We summarize the above result in the form of the following Theorem 8.1. Theorem 8.1. Let N ∗ be the optimal threshold value of ‘N’ that minimizes the average cost per unit time under the linear cost structure CA (N ), then N ∗ of an M x /G/1 queue with second optional service and unreliable server under N-policy is given by

  k −1 − V (ε1 , ε2 , p)  (k − j)ϕj > N = min k ≥ 1  ;  j=0 Ch 

∗

(1)

(1)

where V (α1 , α2 ) = λCs [1 − λa[1] {β1 (1 + ε1 (g1

+ d(11) )) + β2(1) (1 + ε2 (g2(1) + d(21) )) + pγ (1) }].



Remark 8.1. It Note that the optimal threshold value of ‘N’ can be determined from Theorem 8.1 by selecting the best positive value of ‘k’, which is one of the integers surrounding ‘N’. It should be noted here that the optimal threshold value should be 1 if Ch Cs

>

λ⌊1 − λa[1] {β1(1) (1 + ε1 (g1(1) + d(11) )) + β2(1) (1 + ε2 (g2(1) + d(21) )) + pγ (1) }⌋ .  k∑ −1 (k − j)ϕj j =0

9. Numerical illustration This section contains some numerical illustrative examples. Note first that the distributions of the service times, the delay times, the repair times, and the vacation times are not required in order to derive the optimal value of the threshold level N. Only the first and second moments of these distributions are required. Nevertheless, we will assume that the second moments of these distributions are equal to twice the square of the first moments, as in the exponential distribution. The distribution of the sizes of successive arriving batches is required though, and we will assume that it is the geometric distribution with parameter α , so that a[1] = 1/α and a[2] = (2 − α)/α 2 . We note that, in this case, it is straightforward to show that φn = α for all n = 1, 2, . . . , N − 1, and thus L(N − 1) = 1 + N α and M (N − 1) = N (N − 1)α/2. The values assumed for the non-monetary system parameters are summarized in Table 1. For the monetary system parameters, the unit costs are summarized in Table 2: Fig. 1 shows the variations of the average cost per unit of time as the threshold level varies. The graph is convex. The optimal value of the threshold level is N ∗ = 6. This means that if the server finds fewer than 6 customers in the queue at a service completion epoch, then he should remain idle and wait for the queue size to become 6. When it does, then the busy period starts. In the case when there are more than 6 customers in the queue at a service completion epoch, then the server resumes service with the next customer in line. The optimal cost of this policy is CA (N ∗ ) = 34.6201. For this case, we also used expression (7.1) and (7.2) to find that the steady-state availability of the server is Aν = 95.41% and the failure frequency of the server is Wf = 9%. Note that the system utilization factor is ρH = 0.246.

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G. Choudhury, L. Tadj / Mathematical and Computer Modelling 54 (2011) 673–688 Table 1 Data (base values) for numerical example. Processes

Parameter

Parameter value

Arrival process

Arrival rate Geometric distribution parameter

λ=1 α = 0.5

Service process

FPS

Mean delay time Mean failure rate

β1(1) = 0.05 (1 ) (1) g1 = β1 /5 (1 ) (1) d1 = g1 /10 ε1 = 0.5 β2(1) = 0.05 (1 ) (1) g2 = β2 /5 (1 ) (1) d2 = g2 /10 ε2 = 0.5

Probability of a vacation Mean vacation time

p = 0.5 γ (1) = 0.45

Mean service time Mean repair time Mean delay time Mean failure rate

SPS

Mean service time Mean repair time

Vacation process

Table 2 Unit costs for numerical example. Unit cost

Unit cost value

Startup cost per cycle Holding cost per unit of time Operating cost per unit of time

Cs = 100 Ch = 5 Cw = 10

Table 3 Effect of the FPS breakdown rate ε1 on the optimal threshold and optimal cost.

ε1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

N∗ CA (N ∗ )

6 34.6166

6 34.6175

6 34.6184

6 34.6193

6 34.6201

6 34.6210

6 34.6219

6 34.6228

6 34.6237

Fig. 1. Average cost vs threshold level.

The effect of the system parameters on the optimal policy can be easily undertaken numerically. For example, Tables 3–6 show the effect on the optimal threshold level N ∗ and on the optimal cost CA (N ∗ ) of the parameters of interest to us in this paper, namely, the breakdown rates ε1 and ε2 , the parameter of the geometric distribution α , and the probability of the Bernoulli vacation p, respectively. All the other parameters are kept unchanged. We see that the parameter of the geometric distribution has the greatest impact on the optimal policy. It is the only one that causes a change in the optimal threshold level N ∗ . It also causes the greatest change in the value of the objective

G. Choudhury, L. Tadj / Mathematical and Computer Modelling 54 (2011) 673–688

687

Table 4 Effect of the SPS breakdown rate ε2 on the optimal threshold and optimal cost.

ε2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

N∗ CA (N ∗ )

6 34.6183

6 34.6187

6 34.6192

6 34.6197

6 34.6201

6 34.6202

6 34.6211

6 34.6215

6 34.6220

Table 5 Effect of the arriving batch sizes on the optimal threshold and optimal cost.

α ∗

N C A (N ∗ )

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

— —

5 90.2835

6 40.0043

6 39.0250

6 34.6201

6 31.7244

6 29.5806

5 27.8873

5 26.3718

Table 6 Effect of the Bernoulli vacation on the optimal threshold and optimal cost. p ∗

N CA (N ∗ )

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

6 34.4030

6 34.4523

6 34.5048

6 34.5607

6 34.6201

6 34.6831

6 34.7499

6 34.8204

6 34.8951

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