bulk queueing system with multiple vacations, n-policy, balking and ...

International Journal of Pure and Applied Mathematics Volume 115 No. 9 2017, 459-469 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue

BULK QUEUEING SYSTEM WITH MULTIPLE VACATIONS, N-POLICY, BALKING AND CONTROL POLICY ON REQUEST FOR RE-SERVICE S. Sasikala1∗ , K. Indhira2 , V.M. Chandrasekaran3 1,2,3 School of Advanced Sciences, VIT University, Vellore, Tamil Nadu-632014. 1 [email protected] 2 [email protected] 3 [email protected] Abstract In this paper, we have discussed the steady state behaviour of M X /GK /1 queueing system with control policy on request for re-service, N-policy, balking and multiple vacations. Customers arrive according to compound Poisson process and they may balk with probability b. The server provides service by fixed bulk bulk service rule. Re-service rendered once and when the length of the queue is less than K. On completion of a vacation, if the length of the queue is fewer than N , then the server takes one more vacation. This process continues till the length of the queue reaches atleast N . After a re-service, the server either takes a vacation or starts to provide service depends on the length of the queue. The probability generating functions of various completion epochs, queue length and system length are

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obtained. Further, some important performance measures presented for the proposed queueing system. AMS Subject Classification: 60K25, 90B22, 68M20 Key Words: Fixed bulk service rule, re-service, multiple vacations, N-policy.

1

Introduction

Bulk queues are common in many real-life situations such as an awesome park, giant wheel, manufacturing process, etc. Bulk service queues originated with Bailey [2]. Chaudhry and Templeton [3] have presented the elaborate work on bulk queues. Vacation queueing models are the efficient way to use the idle time for useful internal work. Doshi [4] investigated an exhaustive work on vacation queueing models. Jau-Chuan Ke [7] derived the important characteristics of M X /G/1 queue with balking and variant vacation policy. An M X /G(a,b) /1 queue with multiple vacations and N-policy analyzed by various authors such as Arumuganathan and Jeyakumar [1]. Lee et al. [6] have studied the fixed bulk service queueing system with single and multiple vacations. Performance measures for M X /G(a,b) /1 queue with multiple vacations and controlled re-service analyzed by Jeyakumar and Arumuganathan [5]. This paper deals with an M X /GK /1 queue with balking, multiple vacations and N -policy. Customers arrive in batches with rate λ. The server always serves fixed number of customers in a batch, say K. If the queue size is less than K at the end of a service and there is a request for re-service from the leaving batch, then the server provides re-service once. If there is no request for re-service, then the server follows a vacation with probability 1 − π. If the queue size is greater than or equal to K at the end of a service the server resumes its service. Upon returns from a vacation, if the length of the queue is less than N , then the server takes one more vacation. This process continues till the length of the queue reaches atleast N . After re-service the server either takes a vacation or starts to provide service depends on the length of the queue depends on the length of the queue. On completion of re-service, if the queue size is less than K, then server takes a vacation else resumes its service. This paper organized as follows: Section 1 contains an intro-

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duction. Section 2 presents the brief description of the proposed queueing system. Steady state equations for the queueing system presented in section 3. In section 4, we have given probability generating functions of various completion epochs, system length and queue length. Further, some important performance measures for this queueing system derived in section 5. Conclusions inclined in section 6.

2

Model description

A real life situation exists in cake manufacturing industry. Let C be the batch size random variable for arrival, χm be the probability that m customers arrive in a batch and C(z) be its probability generating function. Let B(x), V (x) and R(x) be the cumulative distribution functions for service time, vacation time and re-service time. Further, b(x), v(x) and r(x) be their respective probability density functions. We assume that at an arbitrary time t, B 0 (t), V 0 (t) and R0 (t) denotes the remaining service time, remaining vacation time and remaining re-service time. Denote the Laplace transforms of b(x), v(x) and r(x) as B ∗ (θ), V ∗ (θ) and R∗ (θ). Let us assume that Qq (t), Qs (t) and Q(t) be the number of customers in the queue, number of customers in the service and number of customers in the system. We further define, Z(t) = j, the server is on j th vacation and the random variable   0 − if the server is busy Y (t) = 1 − if the server is on re-service   2 − if the server is on vacation The supplementary variables are introduced to obtain bivariate Markov process{Q(t), Y (t)}. QK,n (x, t)dt = P {Qs (t) = K, Qq (t) = n, x < B 0 (t) ≤ x + dt, Y (t) = 0} Pj,n (x, t)dt = P {Qq (t) = n, x < V 0 (t) ≤ x + dt, Y (t) = 2, Z(t) = j}, j ≥ 1 Gn (x, t)dt = P {Qq (t) = n, x < R0 (t) ≤ x + dt, Y (t) = 1}

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Steady-state system equations

The following steady state differential-difference equations are obtained by using the method of supplementary variables which was introduced by Cox [2]. 0

−QK,0 (x) = −λQK,0 (x) + λbQK,0 (x) + QK,K (0)b(x) + GK (0)b(x)

0

−QK,n (x) = −λQK,n (x) + λbQK,n (x) + (1 − b)

n X

QK,n−m (x)λχm +

n X

QK,n−m (x)λχm +

m=1

(1)

(2)

GK+n (0)b(x) + QK,K+n (0)b(x), 1 ≤ n ≤ N − K − 1 0

−QK,n (x) = −λQK,n (x) + λbQK,n (x) + (1 − b)

m=1

(3)

GK+n (0)b(x) + Pj,K+n (0)b(x) + QK,K+n (0)b(x), n ≥ N − K 0

−P1,0 (x) = −λP1,0 (x) + λbP1,0 (x) + (1 − π)QK,0 (0)v(x) + G0 (0)v(x)

(4)

0

−P1,n (x) = −λP1,n (x) + λbP1,n (x) + (1 − π)QK,n (0)v(x) + Gn (0)v(x) n X +(1 − b) P1,n−m (x)λχm , 1 ≤ n ≤ K − 1

(5)

m=1

0

−P1,n (x) = −λP1,n (x) + (1 − b)

n X

m=1

P1,n−m (x)λχm + λbP1,n (x), n ≥ K

0

−Pj,0 (x) = −λPj,0 (x) + λbPj,0 (x) + Pj−1,0 (0)v(x), j ≥ 2

0

−Pj,n (x) = −λPj,n (x) + λbPj,n (x) + (1 − b)

n X

Pj,n−m (x)λχm

n X

Pj,n−m (x)λχm , n ≥ N

m=1

(6)

(7)

(8)

+Pj−1,n (0)v(x), 1 ≤ n ≤ N − 1 0

−Pj,n (x) = −λPj,n (x) + λbPj,n (x) + (1 − b)

m=1

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0

−G0 (x) = −λG0 (x) + λbG0 (x) + πQK,0 (0)r(x)

0

−Gn (x) = −λGn (x) + λbGn (x) + (1 − b)

(10)

n X

Gn−m (x)λχm

n X

Gn−m (x)λχm , n ≥ K

(11)

m=1

+πQK,n (0)r(x), 1 ≤ n ≤ K − 1 0

−Gn (x) = −λGn (x) + λbGn (x) + (1 − b)

m=1

(12)

We are defining the following probability generating functions to find the probability generating functions of various completion epochs, queue length and system length, Q∗K (z, θ) =

∞ X

Q∗K,n (θ)z n QK (z, 0) =

n=0

Pj∗ (z, θ) =

∞ X

G∗ (z, θ) =

QK,n (0)z n

n=0

∗ Pj,n (θ)z n Pj (z, 0) =

∞ X

Pj,n (0)z n

n=0

n=0 ∞ X

∞ X

G∗n (θ)z n G(z, 0) =

n=0

∞ X

Gn (0)z n

n=0

By taking Laplace-Stieltjes transform on both sides of the Eqs. (1) - (12) and by using the above probability generating functions, we obtained as Q∗K (z, θ) =

(B ∗ (λ − λb − (1 − b)λC(z)) − B ∗ (θ)) f (z) (θ − λ + λb + (1 − b)λC(z)) (z K − B ∗ (λ − λb − (1 − b)λC(z))) (13)

f (z) = G(z, 0)+Pj (z, 0)−

K−1 X n=0

∗

P1∗ (z, θ) =

QK,n (0)z n − ∗

K−1 X n=0

Gn (0)z n −

(V (λ − λb − (1 − b)λC(z)) − V (θ))  K−1 K−1 P P n n (1 − π) QK,n (0)z + Gn (0)z n=0

n=0

(θ − λ + λb + (1 − b)λC(z))

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Pj,n (0)z n

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Pj∗ (z, θ)

=

∗

G (z, θ) =

4

(V ∗ (λ − λb − (1 − b)λC(z)) − V ∗ (θ))

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NP −1

Pj−1,n (0)z n

n=0

(15)

(θ − λ + λb + (1 − b)λC(z)) π (R∗ (λ − λb − (1 − b)λC(z)) − R∗ (θ))

K−1 P

QK,n (0)z n

n=0

(16)

(θ − λ + λb + (1 − b)λC(z))

Probability generating functions

Probability generating functions of various completion epochs, system length and queue length obtained as follows

4.1

Probability generating function of queue size at service completion epoch

QK (z) =

(B ∗ (λ − λb − (1 − b)λC(z)) − 1) f (z) (−λ + λb + (1 − b)λC(z)) (z K − B ∗ (λ − λb − (1 − b)λC(z)))

f (z) = πR∗ (λ − λb − (1 − b)λC(z)) (1 − π)

4.2

Pj (z) =

K−1 X i=0

i

qi z +

K−1 X i=0

i

gi z +

N −1 X i=0

pi z

i

K−1 X i=0

!

−

(17)

qi z i + V ∗ (λ − λb − (1 − b)λC(z)) K−1 X

i

qi z +

i=0

K−1 X

i

gi z +

i=0

N −1 X

pi z

i

i=0

!

Probability generating function of queue size at vacation completion epoch   K−1 K−1 NP −1 P P (V ∗ (λ − λb − (1 − b)λC(z)) − 1) (1 − π) qi z i + gi z i + pi z i i=0

(−λ + λb + (1 − b)λC(z))

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4.3

G(z) =

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Probability generating function of queue size at re-service completion epoch π (R∗ (λ − λb − (1 − b)λC(z)) − 1) (−λ + λb + (1 − b)λC(z))

K−1 P

qi z i

i=0

(19)

Probability generating function of queue length

Let Pq (z) be the probability generating function (PGF) of the queue size, then Pq (z) = QK (z) + Pj (z) + G(z) =

N rq (z) Dr(z)

(20)

N rq (z) = π(z K − 1) (R∗ (λ − λb − (1 − b)λC(z)) − V ∗ (λ − λb − (1 − b)λC(z))) K−1 X i=0

qi z i + (z K − 1) (V ∗ (λ − λb − (1 − b)λC(z)) − 1) "K−1 X i=0

i

qi z +

K−1 X

i

gi z +

i=0

Dr(z) = (−λ + λb + (1 − b)λC(z)) z K − B ∗ (λ − λb − (1 − b)λC(z))

4.5

N −1 X




pi z

i

i=0

#

Probability generating function of system length

Let Ps (z) be the probability generating function (PGF) of the number of customers in system at arbitrary time epoch, then Ps (z) = z K QK (z) + Pj (z) + G(z) =

N rs (z) Dr(z)

(21)

 N rs (z) = π(z K − 1)B ∗ (λ − λb − (1 − b)λC(z)) R∗ (λ − λb − (1 − b)λC(z))−

 K−1 X V ∗ (λ − λb − (1 − b)λC(z)) qi z i + i=0

(z K − 1)B ∗ (λ − λb − (1 − b)λC(z)) (V ∗ (λ − λb − (1 − b)λC(z)) − 1) K−1 N −1 h K−1 i X X X qi z i + gi z i + pi z i i=0

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Some important performance measures

5.1

Expected number of customers in the queue 000

Lq =

00

00

000

Nq (1)Dr (1) − N rq (1)Dr (1) 3Dr00 (1)2

(22)

Where, Dr00 (1) = 2λ(1 − b)E(X)(K − λ(1 − b)E(X)E(B))
Dr000 (1) = 3λ(1 − b)E(X 2 ) K − λ(1 − b)E(X)E(B) + 3λ(1 − b)E(X)
K(K − 1) − λ(1 − b)E(X 2 )E(B) − λ2 (1 − b)2 E(X)2 E(B 2 ) N rq00 (1) = 2λKπ(1 − b)E(C)E(R)

K−1 X i=0

K−1 X  qi qi + 2λK(1 − b)E(C)E(V ) (1 − π) i=0

+

K−1 X

gi +

N −1 X

pi

i=0

i=0



X
K−1 qi N rq000 (1) = 2πλ(1 − b)K(K − 1)E(C) E(R) − E(V ) i=0

  +3λπK(1 − b) E(C 2 )(E(R) − E(V )) + λ(1 − b)E(C)2 (E(R2 ) − E(V 2 )) 6λπK(1 − b)E(C)(E(R) − E(V )) 3λ(1 − b)K(K − 1)E(C)E(V )

X  K−1

5.2

qi +

i=0

K−1 X i=0

gi +

iqi +

K−1 X i=0

K−1 X i=0

K−1 X

iqi +

i=0

N −1 X i=0

gi +

qi +

i=0

N −1 X i=0

igi +

 pi +

 pi +

N −1 X

ipi

i=0

Expected number of customers in the system 000

Ls =



K−1 X

K−1 X i=0

i=0


 3λK(1 − b) E(C 2 )E(V ) + λ(1 − b)E(C)2 E(V 2 ) 6λK(1 − b)E(C)E(V )

qi +

K−1 X

00

00

000

N rs (1)Dr (1) − N rs (1)Dr (1)

(23)

2

3Dr00 (1)

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N rs00 (1) = λπK(1 − b)E(C)E(R)

K−1 X i=0

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qi + λK(1 − b)E(C)E(V )

K−1 K−1 N −1 X X X   (1 − π) qi + gi + pi i=0

i=0

i=0

N rs000 (1) = 6πKλ2 (1 − b)2 E(B)E(C)2 E(R) − E(V )

X
K−1

qi +

i=0

X   K−1 3πKλ(1 − b) E(C 2 )(E(R) − E(V )) + λ(1 − b)E(C)2 (E(R2 ) − E(V 2 )) qi + i=0


6πKλ(1 − b)E(C) E(R) − E(V )

3πK(K − 1)λ(1 − b)E(C) E(R) − E(V )

K−1 X

iqi +

i=0

X
K−1

qi +

i=0

N −1 K−1 X X X   K−1 pi + gi + qi + 3K(K − 1)λ(1 − b)E(C)E(V )

 6Kλ2 (1 − b)2 E(C)2 E(B)E(V )

i=0

i=0

i=0

K−1 X

K−1 X

N −1 X

qi +

i=0

3λK(1 − b)E(C 2 )E(V ) N −1 X i=0

5.3

gi +

i=0

i=0



K−1 X

qi +

i=0

 pi +

K−1 X

gi +

i=0

N −1 K−1 X X X    K−1 pi + gi + qi + pi + 3λ2 K(1 − b)2 E(C)2 E(V 2 )

 6λK(1 − b)E(C)E(V )

i=0

i=0

i=0

i=0

K−1 X

iqi +

K−1 X

igi +

i=0

N −1 X

ipi

i=0

Expected waiting time in the queue

Expected waiting time in the queue obtained by using Littles formula Wq = Lq /λE(X) , where Lq is given in Eq. (22).

5.4

Expected waiting time in the system

Expected waiting time in the system obtained by using Littles formula Ws = Ls /λE(X), where Ls is given in Eq. (23).

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6

Conclusion

In this work, we have examined the M X /GK /1 queueing system with control policy on request for re-service, N-policy, balking and multiple vacations. Steady-state system equations are obtained. We have obtained the probability generating functions of various completion epochs, queue length and system length by using the method of supplementary variables. Further, we have given some important performance measures for the proposed queueing system.

References [1] R. Arumuganathan, S. Jeyakumar, Steady state analysis of a bulk queue with multiple vacations, setup times with Npolicy and closedown times, Applied Mathematical Modelling, 29 (2005), 972-986. [2] N. T. Bailey, On queueing processes with bulk service, Journal of the Royal Statistical Society. Series B (Methodological), (1954), 80-87. [3] M. Chaudhry, J. Templeton, A first course in bulk queues, John Wiley & Sons (1983). [4] B. T. Doshi, Queueing systems with vacations - survey, Queueing systems, 1 (1986), 29-66. [5] S. Jeyakumar, R. Arumuganathan, A non-markovian bulk queue with multiple vacations and control policy on request for re-service, Quality Technology and Quantitative Management, 8 (2011), 253-269. [6] H. W. Lee, S. S. Lee, K. Chae, A fixed-size batch service queue with vacations, International Journal of Stochastic Analysis, 9 (1996), 205-219. [7] Jau-Chuan. Ke, Operating characteristic analysis on the M X /G/1 system with a variant vacation policy and balking, Applied Mathematical Modelling, 31 (2007), 1321-1337.

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