A Vacation Queueing Model with Bulk Service Rule

ISSN 2350-1049
International Journal of Recent Research in Interdisciplinary Sciences (IJRRIS)
Vol. 3, Issue 1, pp: (1-12), Month: January - March 2016, Available at: www.paperpublications.org
Page | 1
Paper Publications
A Vacation Queueing Model with Bulk Service
Rule
1
PUKAZHENTHI.N, 2
EZHILVANAN.M
Department of Statistics, Annamalai University, Annamalai Nagar - 608 002
Abstract: The present paper analyzes a vacation queueing model with bulk service rule. Considering distribution of
the system occupancy just before and after the departure epochs using discrete-time analysis. The inter-arrival
times of customers are assumed to be independent and geometrically distributed. The server works at different
rate, rather than completely stopping service vacations. The service times during a service period and vacation
time geometric distribution. Some performance measures and a cost model have been presented and an optimum
service rate has been obtained using convex function. Numerical results showing the effect of the parameters of the
model on the key performance are presented.
Keywords: Batch service, Discrete-time queue, Departure epochs, slot, system occupancy and inter departure time
distribution, vacation termination epochs.
1. INTRODUCTION
The article deals with a vacation queueing model with bulk service rule .Using the discrete time analysis of a queue of
packets of information in telecommunication networks, has been mode in the study of transportation systems. These
queueing of packets can be model even when time is divided into equal intervals called slots. The numbers of customers
that arrive one by one according to a follow geometric distribution, arriving packets first joint and wait in the queue called
secondary queue before and move in batches of packets of minimum size number L and maximum size number K to the
queue called primary queue after whenever the latter opens and service times of all customers are assumed to be a
common follows geometric distribution. The ongoing service when ( ) and the service times are not affected by
such inclusion.
In the queueing literature, exposes many useful results for queues with bulk service of a single server and queues with
vacation. Relatively few results are available for queues that allow before and after both customer and sever vacation. A n
extensive investigation on single server queueing system with vacation has been mode in recent years by many
researchers. Miller (1964) is the first to study such a model, where the server is unavailable during some random length of
time (referred to vacation) for the queueing system. Queue with vacation has been studied by number of authors
Cooper (1970) and Doshi (1986). Harris and Marchal (1988) and Shanthikumar (1988) have proved the stochastic
decomposition result for unfinished work in the system and additional delay due to the vacation time respectively in more
general setting.
Krishna Reddy.et.al., (1992) analyzes a queue with general bulk service and server vacation. Dhas.et.al., (1992) considers
a pre-emptive priority queue with general bulk service and vacation. Lee et.al (1996) derives the probability generating
function of a fixed size batch service queue with server vacation. Lee. et.al., (1997) establishes for the first time in their
on batch service queueing system with vacation sever unavailability may reduce the waiting time in some batch service
queue system. Kalyanaraman and Ayyapan (2000) analysed a single server vacation queue with balking and with single
and batch service. They derived the steady probability distribution of the number of customers in the queue. Chinnadurai.
et.al.,(2002) analyses the vacation queue with balking and with state dependent service rate.

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VeenaGoswami and VijiyaLaxmi (2011) discussed the inter-arrival times of customers are arbitrarily distributed, the
service times and vacation times are both geometrically distributed. The server starts service in batches of maximum
number size 'b' and minimum number threshold value 'a', otherwise if goes for vacation. There is a provision to include
the late arriving customer in the ongoing service batch if the size of the batch being served less than a threshold value
( ) and the service time is not affected by such inclusions. Recently, VijiyaLamxiet.al., (2013) studies
discrete-time single server queue and the inter-arrival times of customer are assumed to be (i.i.d).The service time and
service time during a vacation period follows geometrically distributed. Sree Parimalaand Palaniammal (2014) present the
analytical solution of two servers bulk service queueing models in which only one server is allowed for vacation at a time
to avoid the inconvenience to the customers.
In the analysis of the present study is continued under the assumption that service time of batches are independent of the
number of packets in any batch and the simultaneous occurrence of a packet arriving and a vacation termination in a
single slot is ruled out. The primary focus is on deriving various performance measures of the steady state distribution of
the batch server at departure instants and also on numerical illustrations.
In the present investigation an attempt has been made to analyze the server’s delayed and single vacation. This paper is
organized as follows: Section 2 contains a brief motivation and description. In Section 3, the joint distribution of system
occupancy at departure epoch is derived and inter-departure time and number of packets are discussed in Section 4.
Numerical illustration in the form of tables and graphs are presented in Section 5. Finally, in section 6 concludes the
paper.
2. MOTIVATION AND DESCRIPTION OF THE MODEL
Queueing system in which the server works on primary and secondary customers arise naturally (and sometimes not so
naturally) as model of many computer, communication and production systems. The server working on the secondary
customers is equivalent to the server taking a vacation and not being available to the primary custom as during this period.
Thus, there is a natural interest in the study of queueing system with server vacation.
2.1. The arrival time distribution:
In this model geometric distribution and single vacation queue are considered. For geometric vacation the following
mathematical definition is adopted. The server takes a vacation when the system becomes empty after a service or a
vacation. However after each service ,if there are some customers in the queue the server may take a vacation with some
probability 'p' or start a new service (if a customer is present) with probability ( ). In this model we assume that
the inter-arrival "An" of customers are independent and geometric distribution with probability mass function (p.m.f).
* ( ) ( ) + i.e.
( ) ( ) ( )
2.2. The mean number of customers in the inter-arrival time distribution:
The probability generating function for the number of customers in the system are obtained. The mean and variance of the
number of customer in the system, waiting time of a customer in the system are also obtained. Some particular model are
analyzed by assuming specific probability distribution to service time and vacation time
The probability generating function (p.g.f)
( ) ∑ ( ) ∑ ∑( )
( )
The first order derivative,
( ) ( )
( )

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The second order derivative,
( )
( )
( )
Hence,
( ) ( )
2.3. The variance of number of customers in the inter-arrival distribution:
From the previous calculation
( )
Hence,
( ) ( )
2.4. The service time distribution:
The service time and vacation time are independently and identically distributed random variables. The customer are
served in the order of their arrival in batches of size, lying in the closed interval (L,K). The server takes a vacation, if the
number of customers waiting in queue is less than number of customers to wait in queue. when the server returns from
vacation, if more then (L-1) or less than (K+1) customers are waiting then all the customers are taken into service.
Customers, who arrive, while a group of customers are being served, cannot join the ongoing service even if some space
is available. The service time for the batch is of length , where the are random variables which follows
geometric distribution with (p. m. f). * ( ) ( ) + service time is independent of the number of
slots in the batch.
( ) ( ) ( )
2.5. The mean number of customers in the service time distribution:
The probability generating function of service time distribution B (z) is given by
( ) ∑ ( ) ∑( ) ∑( )
which implies that
( )
( )
The first order derivatives,
( ) ( )
and second order derivatives,
( ) ( )
Hence,
( ) ( )

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2.6. The variance of number of customers in the service time distribution:
From the above result, variance can be write as
( ) ( ) ( )
3. QUEUEING SYSTEM UTILIZATION
Utilization is the proportion of the system resources which is used by the traffic when arrives at it (Geometric arrivals and
Geometric server), then it is given by the mean arrival rate over the mean service rate that is,
( )
( )
( )
The server load,
( )
which is less than 1.
The probability of a packet arriving at the close of the slot is ' and it is not arriving at that point is which implies that
the inter-arrival time geometrically distributed. A batch of packets of size ( ) starts service at the beginning of
a slot, and may only end service just before the end of a slot. The probability of a batch being transported by the server at
the end of a slot is probability and the probability of the batch not being transported is The probability that the
server terminates its vacation when it is in vacation at the close of a slot is and that of it not terminating the vacation
at that point is which implies that the vacation time is geometrically distributed. Service time of a batch is
independent of the number of packets in a batch and simultaneous occurrence of both arrival and departure the vacation
termination is a single slot are ruled out.
4. THE JOINT DISTRIBUTION OF SYSTEM OCCUPANCY AT DEPARTURE EPOCHS
Let * + be the number of packets accumulated in the system queue and service just after the server has left
with batch. The steady state distribution * ( ) ( ) ( ) + of system
occupancy at departure epochs is derived using the embedded Markov Chain technique.
Let be random variable denoting the number of packets that reach the system during the nth
service. Then the joint
distribution* ( ) + of can be derived as follows:
( ) ∑ ( ) * +* + ( )
For
( ) ( ) ( )
Putting so that . Further on the limit and implie .
Then the above expression will be
( ) ∑ ( ) * ( ) +* ( ) +
( )
( ( )( ))
∑ ( ) ( )⁄

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( )
. (( )( ))/
( )
( ( ))
( )
( )
( )
( ) ( )
Let be random variable denoting the number of packets that reaching the system during a vacation period. Then the
joint distribution *( ) + of can be derived as follows:
( ) ∑ ( ) * +* + ( )
For
( ) ( ) ( )
Putting so that . Further on the limit and implies . Then the
above expression will be
( ) ∑ ( ) * ( ) +* ( ) +
( ) * ( ) + ∑ ( ) * ( ) ( ) +
( )
( ( )( ))
∑ ( ) ( )⁄
( )
. (( )( ))/
( )
( ( ))
( )
( )
( )
( ) ( )
Let consider the basic stochastic process * + which is a Markov Chain (MC) induced by the MC related to the
queue. The transition probability matrix of the Marko Chain is ( ).
where
{
∑
∑ ∑
( ) ( ) ( )
( )
Let the unknown probability row vector ( ) represent solving the following system of equation.
( )
where denotes the stands for the row vector of unit elements. A number of numerical using the method could be
suggested to solve the system of equation (14). For example, an algorithm for solving (14) is given by Latouche and
Ramaswami (1999).

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5. DISTRIBUTION OF OCCUPANCY JUST BEFORE BATCH DEPARTURE EPOCHS
The joint probability ( ) ( ) say that packets arrive during a vacation period cosisting of slot
is
( ) ( ) * ( ) +* ( ) + ( )
Let the partial generating function of the vacation period, given that packets arrive during this vacation period be
( )
( ) ∑ ( ) ∑ ( ) * ( ) +* ( ) +
( ) ( ) ( )
It is observed that the following results can be obtained from (16) for
( ) ( )
∑ ( ) ( ) ( )
which is the probability generating function of * ( ) +
( ) ( ) ( ) ( ) ( ) ( )
( )
( )( ) ( )* +
( )
( ) ( )
Let random system occupancy immediately at the beginning of the service period. As the respective distribution of
is* ( ) ( )+ +,it could be seen that* ( ) ( ) ( )+
+, which is
( ) ( )
{
( ) ∑
( ) ∑ ( ) ∑( )
( )
6. INTER-DEPARTURE TIME AND NUMBER OF PACKETS IN A BATCH
Let random varible denote the time number of slots between the departures of two consecutive batches of packets. The
inter-departure time consists of the inter-arrival time of the (L-X) packets needed to make up a batch and the service time
of that batch. Hence, the distribution * ( ) ( )+ of inter-departure time is
( ) ( ) ∑ ( ) ∑ ,∑ ( ) ( ) ∑ ( ) ( )
( ) ( )- ( ) ( )
where * stands for convolution operation and denotes i-fold convolution of a(k) with itself . The probability generating
function of is
( ) ( ) *∑ ( ) ∑ ,∑ ( ) ∑ ( )* ( )+( )
- ( )+

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The first order derivative,
( ) ( ) *∑ ( ) ∑ ,∑ ( ) ∑ ( ) * ( )+( )
- ( )+
( ) *∑ ,∑ ( ) ∑{ ( )}* ( )+( )
( ) ( )* ( )+( ) ( )
- ( )+
The second derivative,
( ) ( ) *∑ ( ) ∑ ,∑ ( ) ∑ ( ) * ( )+( )
- ( )+
( ) *∑ ,∑ ( ) ∑{ ( )* ( )+( ) ( )( )* ( )+( ) ( )}- ( )+
( ) *∑ ,∑ ( )
∑ { ( )* ( )+( )
( )( )( ( ))
( ) ( )
( )( )( ( ))
( ) ( )
( )( ) {( ) ( )
( )( ( )) ( )( )
( )}}- ( )+
( ) *∑ ,∑ ( ) ∑{ ( )* ( )+( )
( )( )* ( )+( )
( )}- ( )+
and thus
( ) ( ) *∑ , ( ) ∑ ( )( ) ( )
- ( )+
( ) ( ) ∑ ( ) ∑ *,( ) ( ) ( ) ∑ ( ) ( ) ( ) ( )- ( )+
where
( ) ( )
( ) *∑ * ( )
∑ { ( )( ) ( ) ( )( )*( ) ( ) ( )+ ( )( ) ( )
( )( )*( ) ( ) ( )+}+ ( )+
Let S be a random variable denoting the number of lots in a batch at the beginning of a batch service and
( ) ( )for

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( )
{
( )
( )
{
( ) ∑ ( ) ∑ ( )
( ) ∑ ( ) ( )
∑ ( ) ∑ ( ) ∑ ( )
∑ ( )
The joint distribution * ( ) ( ) + and S is
( ) ( ) [ ( ) ∑ ( ) ∑ ( ) ∑ ( )]
( ) [ ( ) ( ) ∑ ( ) { ( ) ∑ ( ) ( )}]
, ( ) ( )- [ ( ) ( )
∑{ ( )} ( )] ( )
where ( ) when and 0 otherwise
The expectation of the random variables ,S and their product ,S given below.
( ) ( ) ( ) {∑ ( ) ∑ ( )}
∑ ( ) , ( ) ∑{ ( ) ( ) ( )}-
( ∑ ) {∑ ( ) ( )} ( )
In the similar manner the variance of and S can be calculated. To know how strongly these two random
variables depend on each other, co-efficient of correlation can be compute and it can be defined as
( ) ( ) ( )
√ ( )√ ( )
( )
Similarly Let random variables denoting the number of lots in a batch just before the batch * ( )
( ) +
( ) {
∑ ( )
( )
( )
The moment of can be calculated from (22).By using (15) in one could rewrite (16) in terms of g (.) values as we get
( ) ( ) ∑ ( ) ∑[ ( )
( )] ( ) ( ( )) ∑ ( ) ( )

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We derive joint distribution of and the usual arguments lead to the following difference- differential equation
* ( ) ( ) +
( ) ( ) ( ( ) ∑ ( ) ∑[ ( ) ( )] ( )) ( ( ))( ( )) ( ) ( )
Thus,
( ) ( ( ) ∑ ( ) ∑,( ) ( ) ( )- ( )) ( ( ) ( )) ∑ ( ) ( )
As in (17), one can calculate the correlation co-efficient, say , between and .
( ) ( ) ( )
( )
7. NUMERICAL ILLUSTRATION
Consider the values p=0.98, p=0.45 ,p2=0.28, and K=20 be considered so that For
( ) below shows the numerical values of the mean, variance and the co-efficient of correlation r (r2)of the
stationary queue under study. In the results obtained on * ( ) + sum over ‘i’ is truncated at 150 and been
shown in table-1 below.
Table-1: Departure epoch measures
L E(Db) Var(Db) E(S) Var(S) E(S2) V(S2)
1
2
3
4
5
6
7
8
9
0
11
12
13
14
15
16
17
18
19
20
6.077
6.575
7.222
7.974
8.803
9.686
10.607
11.557
12.526
13.509
14.501
15.499
16.498
17.495
18.482
19.449
20.376
21.229
21.945
22.408
10.647
9.102
7.671
6.484
5.554
4.851
4.332
3.957
3.690
3.505
3.382
3.308
3.277
3.291
3.360
3.512
3.794
4.276
5.018
5.898
2.802
3.290
3.923
4.661
5.472
6.338
7.241
8.172
9.122
10.087
11.061
12.043
13.030
14.021
15.014
16.010
17.006
18.004
19.002
20.000
7.518
6.015
4.617
3.453
2.535
1.836
1.315
.932
.653
.453
.309
.208
.136
.087
.053
.031
.016
.007
.002
.000
4.00
4.49
5.12
5.86
6.67
7.53
8.44
9.37
10.32
11.28
12.25
13.23
14.21
15.18
16.15
17.10
18.01
18.84
19.55
20.00
10.05
8.55
7.15
5.99
5.08
4.38
3.86
3.48
3.19
2.97
2.80
2.64
2.48
2.28
2.03
1.68
1.24
.73
.25
.00
.85590
.82839
.79134
.74543
.69172
.63200
.56859
.50408
.44099
.38108
.32563
.27521
.22972
.18883
.15169
.11733
.08521
.05385
.02199
.00000
-.01019
-.00980
-.01025
-.01126
-.01260
-.01408
-.01560
-.01719
-.01910
-.02186
-.02629
-.03371
-.04579
-.06489
-.09349
-.1337
-.18503
-.23976
-.27140
.00000
The r1 and r2 value is zero when L=K= 20 as the batch size is constant in this specific case. For L=1 to 20, values of the
co-efficient of correlation r1 and r2 are negative. It indicates long inter-departure times correspond to small number in
system occupancy at departure epochs to form a batch at the beginning of their service epoch. Mean values of both
random variables Db and var (Db) increase with increase in L. But the variance values of the random variables decrease
with the increase with the increase in L values as the range value between L and K becomes smaller and smaller value.

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7.1. Performance measure and cost function:
Cost function involving random variables and ( ) can be formulated. Different cost values incurred by the
coefficient of variation (CV) of S and by the coefficient of determination (CD) of can be defined as:
Cos due to CV(S) (Cost due to CV ( ))
Cos due to CV( ) (Cost due to CV ( ))
The total expected cost (TEC) function TEC1 and TEC2 say, subject to these costs depend on the parameters of
the batch service system in terms of the pairs of random variables ( )and ( ) and respectively they are
( ) ( )
( ) ( )
TEC1 and TEC2 can be shown as convex functions, as L increases, CV(S) and ( ) decrease and and increase.
Since the closed form of expression is not available for TEC, one can locate only local minimum in specific cases. An
attempt has been made to trace it with the following assigned values:
andK=15so that .
Table - 2: Cost function values
Fig - 1.Graph of TEC1
TEC curve: Db Vs S1 c1=$175.00, c2=$452.00,p=0.950, p1=0.470, α=3 and K=15 so that ρ=0.404255
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10 11 12
TEC1(Db.S1)
L - values
T1
T2
TEC(1)
L c1 CV(S) c2 r1
2
TEC1 c1 CV(S2) c2 r2
2
TEC2
1 19.04 0.04 19.079 106.39 0 106.398
2 9.36 0.17 9.528 85.6 0.03 85.628
3 6.13 0.39 6.519 71.32 0.09 71.406
4 4.52 0.69 5.214 60.75 0.25 61.008
5 3.55 1.09 4.638 52.46 0.63 53.095
6 2.9 1.56 4.462 45.6 1.44 47.031
7 2.43 2.1 4.529 39.6 3.03 42.636
8 2.07 2.67 4.738 34.11 5.95 40.061
9 1.77 3.24 5.009 28.86 10.75 39.606
10 1.51 3.76 5.266 23.67 17.66 41.328
11
12
1.26
1.01
4.21
4.57
5.473
5.573
18.44
13.22
25.86
33.71
44.304
45.925

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An attempt has been made to trace it with the following pre-assigned values: c1=$175.00,c2=$452.00,p=0.950,p1=0.470,
α=3 and K=15 so that ρ=0.404255. Based on the corresponding results of TEC1/TEC2, graphs have been drawn and
presented in Fig-1 and Fig-2 respectively where CV=(S.D/Mean) only.
TEC curve: Db Vs S2: c1=$175.00,c2=$452.00,p=0.950,p1=0.470, α=3 and K=15 so that ρ=0.404255
Fig - 2: Graph of TEC2
The corresponding results of TEC1/TEC2 are presented in table-2 where CV= (S.D/Mean) only. It can be noticed that
local minimum occurs when L = 6 and does when the local minimum is L = 9. Based on this information
regarding local minimum on 'L' ,the facility rendering batch service with the rule (L,K) can call the server (the
transportation vehicle from outside) to come to the service point and thereby saving the amount of transportation cost,
which is the primary objective in transportation and optimization problems.
8. CONCLUSION
The basic analysis relating to the moments of queue contents of a vacation queueing model with bulk service rule is well-
discussed. There are two queues, separated by that arriving packets first join and wait in the secondary queue before and
more in batches of minimum size number L packets and maximum size number K to the primary queue after whenever
the latter opens. This happens at the end of the last slot of a vacation period and is then served in batches ( )
before leaving the primary queue. Both primary and secondary queue have an infinite capacity. Various performance
measures of the steady state distribution of the batch server have been obtained numerically and are provided.
REFERENCES
[1] Chinnadurai,V., Kalyanaraman,R and Ayyappan,G., (2002)." A vacation queue with bulking and with state
dependent service rate", Bulletin of pure and Applied Science, Vol.21E(1),pp,51-57.
[2] Cooper,R.B., (1970)." Queues served in cyclic order: waiting times", Bell System Technical Journal, Vol.49,pp,399-
413.
[3] Dhas, D., Nadarajan,R and Lee,H.W., (1992) " A preemption priority queue with general bulk service and
vacation". Computer in Industry, Vol.18,pp.327-333.
[4] Doshi,B.T., (1986)." Queueing system with vacation server", Queueing Systems, Vol.I,pp.29-66.
[5] Harris,C.M and Marchal,W.G.,(1988)." State dependence in M/G/1 server vacation model". Operation Research.,
Vol.36,pp.560-565.
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9 10 11 12
TEC2(Db,S2)
L values
TE1
TE2
TEC(2)

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[6] Krishna Reddy, G.V., Nadarajan, R. and Kandasamy. P.R., (1992). "A tandem queeue with general bulk service and
server's vacation". Computers in Industry, Vol.20, pp.345-354.
[7] Kalyanaraman. R and Ayyappan. G., (2000). "A single server vacation queue with bulking and with single and batch
service". Bulletin of pure and Applied Science, Vol.19E(2), pp.521-528.
[8] Lee,H.W., Lee,S.S and Chae, K.C., (1996)."Fixed size batch service queue with server vacation", Journal of Applied
Mathematical and stochastic Analysis,Vol.9(2),pp.205-219.
[9] Lee,H.W., Chang,D.I.,Lee,S.S and Chae,K.C., (1997)."Server invariability reduces mean waiting time in some both
service queue system", Computers Operation Reserch.Vol.20, pp.345-351.
[10] Miller, L.W., (1964). "Alternating priorities in multiclass queue", Ph.D., dissertation Cornell University, Ithaca,N.Y.
[11] Shanthikumar, J.G., (1988). "On Stochastic decomposition in the M/H/1 type queue with generalized vacation"
Operation Research, Vol.36, pp.566-569.
[12] Vijaya Laxmi, P. Goswami.V and Jyothsha. K (2013) " Analysis of Discrete- time single Server Queue with Bulking
and multiple working vacations". Quality Technology Quantitative Management. Vol.10, No.4, pp. 443-456.
[13] Veena Goswami. P and Vijaya Laxmi (2011). "Discrete time renewal input multiple vacation queue with accessible
and non-accessible batches". Opsearch ,Vol,48(4),pp.335-334.

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