bulk service queueing model with servers single and delayed vacation

ISSN 2348-5426
International Journal of Advances in Science and Technology (IJAST)
Vol 2 Issue 2 (June 2014)
BULK SERVICE QUEUEING MODEL WITH SERVERS
SINGLE AND DELAYED VACATION
R.Sree parimala1, S.Palaniammal2,
1. Research scholar, 2. Professor& Head,
Department of Science and Humanities
Sri Krishna College of Technology, Kovaipudur, Coimbatore-641 042, Tamil Nadu, India.
E-mail: [email protected],[email protected]
stops, a vacation starts. These predictions help us to
anticipate situations of the system and to take appropriate
measures to shorten the queue. In most of the queueing
models, service begins immediately when the customers
arrives. But some of the physical systems in which idle
servers will leave the system for some other uninterrupted
task referred as vacation. Most of the general bulk service
Queueing models with server vacation have been analyzed
by many authors.
Abstract
This paper deals on a server’s single and delayed vacation
of M/M (a,b)/(2,1) queueing system. In this model it is
assumed that the arrival pattern is Poisson fashion with
parameter λ and service is done in batches which is
exponentially distributed with parameter µ according to the
general bulk service rule introduced by Neuts(9).The
batches are served according to FCFS discipline. The
service starts only when batches of ‘a’ customers are
present. When the queue length is ‘a’ but less than or
equal to ‘b’ then the entire queue is taken up for service. If
there are more than ‘b’ customers in the queue then the
server accepts first ‘b’ customers. In this model the server
takes only one vacation (θ) at a time. (i.e.) on returning
from vacation the server starts serving immediately if there
are ‘a’ customers waiting in the queue. If any one of the
server finds (a-1) customers in the system and other server
is busy or idle, server will stay idle in the system and wait
for the queue size become ‘a’. If the server finds (a-2)
customers in the system and other server is busy or idle, the
server switch over the system and goes for vacation. So in
this system, sever can take only one vacation between two
successive service times. Any one of the server will always
retained in a system.
The steady state solutions and the system
characteristics are derived and analyzed for this model.
Various models studied earlier are discussed as special
cases of our model. The analytical results are numerically
illustrated for different values of the parameters and levels
also.
S.Palaniammal (11 ) has studied M/M(a,b)/(2,1) queueing
model and derived analytic solutions for servers repeated
and single vacation in her Ph.D thesis and presented the
steady state result in terms of characteristic equation of a
difference equation. M.I.Afthab begam (1) has tried analytic
solution for M/M(a,b)/1 queues, Ek/M(a,b)/1 queue with
servers single and multiple vacation in her Ph.D thesis. The
queueing models with vacations have been studied due to
their wide applications in flexible manufacturing or
computer communication systems over more than two
decades. Several surveys on server vacation models have
been done by Doshi (5), Takagi.H(12) analyzed the M/G/1/N
queues with server vacation and exhaustive service., Medhi.J
and Borthakur.A(8) have introduced a general bulk service
rule with two server. Also a bulk queueing model
M/M(a,b,c)/2 with servers vacation has been studied by
Mishra.S.S amd Pandey.N.K (9). The Ek/M(a,b)/1 queueing
system and its numerical results are analyzed by
Chaudry.M.C and Easton.G.D (4). The transient of
Ek/M(a,b)/1/N derived
by
Anjana
solanki
and
Srivastava.P.N(2).
In many waiting line systems, the role of server is
played by mechanical/ electronic device, such as computer,
pallets, ATM, Traffic light, etc., which is subject to
accidental waiting of customers, it may solved by the servers
vacation due to batch criteria. Ke(6) studied the control
policy of the N-Policy M/G/1 queue with server vacations,
startup and breakdowns, where arrival forms a Poisson and
service times are generally distributed.
In the literature described above, customer interarrival times and customer service times are required to
follow certain probability distributions with fixed
parameters.
The present investigation an attempt has been
made to analyze the server’s delayed and single vacation.
The study of queueing model is organized as follows. The
model is described in Section 2. Section 3 provides the
formulation and notations. Steady state behavior of the
system and equation are outlined in Section 4.The steady
Keywords: Delayed vacation, switch over, queue size.
1. Introduction
Queueing models are very useful to provide basic framework
for efficient design and analysis of several practical
situations including various technical systems also
predictions the behavior of system such as waiting times of
customers, various vacations for servers and so forth.
Queueing systems with server vacations have also found
wide applicability in computer and communication network
and several other engineering systems. Such queueing
situations may arise in many real time systems such as
telecommunication, data/voice transmission, manufacturing
system, etc. In computer communication systems, messages
which are to be transmitted could consist of a random
number of packets. Vacation models are explained by their
scheduling disciplines, according to which when a service
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ISSN 2348-5426
International Journal of Advances in Science and Technology (IJAST)
Vol 2 Issue 2 (June 2014)
state solutions have been obtained in Section 5. The
performance measures and mean queue length are derived in
Section 6. The cost of our model are deduced in Section
7.To validate the analytical results and to facilitate the
sensitivity analysis, we present some numerical results for
system performance indices in Section 8 and some
concluding remarks and notable features of investigation
done are highlighted in Section 9.
P0n =
, P1n(t) =
P2n(t) =
and
exists
4. Steady state equations
The steady state equations satisfied by Pjn and Qjn are given
by
(1)
2. Model Description
The study focused on server’s single and delayed
vacation of M/M(a,b)/(2,1) queueing system with switch
over state of server. In this model it is assumed that the
arrival pattern is according Poisson process with parameter λ
and service is done in a batch which is exponentially
distributed with parameter µ.The service starts only when
batches of ‘a’ customers are present. When the queue length
is ‘a’ but less than or equal to ‘b’ then the entire queue is
taken up for service. If there are more than ‘b’ customers in
the queue then the server accepts first ‘b’ customers. In this
model the server takes only one vacation (θ) at a time which
is exponentially distributed. (i.e.) on returning from vacation
the server starts serving immediately if there are ‘a’
customers waiting in the queue. In this model we make the
following assumptions.
(i) If a server finds the other server is on vacation he
will remain in the system, as only one server
is allowed to go on vacation at a time.
(ii) If any one of the server finds (a-1) customers in
the system and other server is busy or idle,
server will stay idle in the system and wait for
the queue size become ‘a’.
(iii) If the server finds (a-2) customers in the system
and other server is busy or idle, the server
switch over the system and goes for a
vacation.
So in this system, sever can take only one vacation
between two successive service times. Any one of the
server will always retained in a system.
(2)
(3)
(4)
(5)
(6)
(7)
3. Mathematical Formulation
The queueing system can be formulated as a
continuous time parameter Markov chain with states Pjn(n≥0,
j = 0,1,2,3) and Qjn ((0 ≤ n ≤ a-2), j = 1,2) denotes the steady
state probabilities, where ‘n’ represents the number of
customers in the queue and ‘j’ signifies the states of the
server. The states of the process
P0n – the probability that one server is idle and the other on
vacation,
P1n – the probability that one server is busy and the other on
vacation,
P2n – the probability that both the servers are busy,
P3n – the probability that one server is busy and the other
switchover from the system
Q1n – the probability that one server is busy and the other
idle,
Q2n – the probability that both are idle in the system
respectively.
we define the following limiting probabilities corresponding
to different states
+
+
(8)
+
+2
(9)
(10)
=
=
(11)
+
(12)
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ISSN 2348-5426
International Journal of Advances in Science and Technology (IJAST)
Vol 2 Issue 2 (June 2014)
=
where
is a constant,
=
(13)
=
,
(x) =
(
) and
+
By adding (2) , (12) and using the equations (1), (11)
(14)
+
5. Computation of steady state solutions:
=
,
Let E denote the forward shifting operator defined
by E(P1n) = P1n+1. From equation ( 7)
from equations (17) and (18) substituting the values of
E
b+1
E+ )
–(
=0
The characteristic equation of the above equation has only
one real root inside the circle |Z| =1 by Rouche’s theorem
when
+
(
is less than 1 then
=
(15)
from equation (9),
Eb+1 – (
E+
)
=
+
)] + k
(
) +
After simplification
(
(
)] + (1+
( ) +
)
( )
=
) +
)
=
=-
the characteristic equation of this equation
has only one real root by Rouche’s theorem which lies in the
interval (0,1) when
+
=
and using equation (15),
+
(1+
)
)
after simplification,
=(
Further giving an expansion and simplifying the above
equation,
(16)
where
+
is a constant and k =
F(
from equation (5), substituting n = a-2, a-3, a-4,...1 and
solving recursively using (15) and (16),
= (
(
) + k
(
) +
=
)
[
+
+
]
here
=
[
( 1-
(19)
and F (
=
)]
(17)
where
–
(
The probability for one of the server is busy and the other
switchover from the system is solved by using (10)
) , R =
=[
and k =
+k
+
]
(20)
Similarly solving equation (13) recursively using (17)
(
)
,
)+k
(
where
)+
(18)
=
and
Also by adding (17), (18) and using the results of equation
(19), we obtain
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ISSN 2348-5426
International Journal of Advances in Science and Technology (IJAST)
Vol 2 Issue 2 (June 2014)
+
=(
+
+F(
normalizing condition. Hence all the probabilities are
completely in terms of the queue parameters.
)
To obtain the value of
(21)
, by using the normalizing
condition
To find the value of constants, from equation (8), using the
results of
,
+
+
+
+k
+
)+
=1
+
(24)
]+
+
]+
+
+
(
)+k
Substituting the results from the equations (19), (22), (15),
(16) and (20) we obtain
k
(
=
)
)]
)+
]+F(
By simplifying and using
(x) =
[
]+
)+
[
]+(
)+
)+
we obtain
(
the value of constant
as follows
+
[
-
-(
(
)+k(
+
)+
. (25)
)
where H(x) =
)]
[
-
]
(22)
6. Performance measures
here
=
-
Also to obtain the value of
[
(
-
[k+
-
]
Performance measures are important features of queueing
systems as they reflect the efficiency of the queueing system
under consideration. The steady-state probabilities at service
completion, vacation termination, departure, and arbitrary
epochs are known, various performance measures of the
queue can be easily obtained such as the average number of
customers in the queue at any arbitrary epoch (Lq),
probability of the servers busy period (
), Probability of
, by adding (4) and (14),
{
)}+F(
-
[
(
) -
]+
one of the servers busy and vacation or idle period (
),
Probability of both the servers vacation or idle period (
),
and Probability of the switch over state to any one of server
(
).
(23)
Mean queue length
where
= [
} - 2
The results of our model are listed below.
]
Let
be the expected number of customers in the queue
then
Thus we obtained all the steady state probabilities in terms
of
which it may now be determined by using the
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ISSN 2348-5426
International Journal of Advances in Science and Technology (IJAST)
Vol 2 Issue 2 (June 2014)
=
+
)+
+
+
+
queueing model.
)+
(26)
7. Cost Model
Using equations (19),(16),(20),(21) and (15) ,
=[
In this section, the cost analysis for the models analyzed by
considering different costs associated with the servers and
customers waiting time. Let
+F(
+
{a +
+k
}+
+
= fixed cost per unit time for each server
+
]
= waiting cost per unit service by each server
(27)
= cost per unit service by each server
here
=
(
+[
= size of the waiting batch in the system
]
If M denotes the expected total cost per unit time for
operating the system, then
)
Probability that both servers are busy (
)
=(
M=2
+
+
where
is the mean queue length and
the probability that the servers busy and
(
+
denotes
represents
Now we present Computational procedures and discussion
of numerical results in this Section. The numerical values of
the performance measures for the various values of the
+F
)
,
),
8. Numerical Analysis
Probability that one server is busy and the other is idle
or on vacation (
)
+
+
the probability of server switch over from the system.
(28)
(29)
5
5.16056
0.5117
0.3115
0.00066
0.000041
8.81293
0.2423
0.6943
0.0158
0.006710
12.1141
23.5442
0.1048
0.0497
0.6796
0.9205
0.1685
0.2292
0.088760
0.014329
6
9.1025
0.7774
0.2701
0.0009
0.000080
12
11.2353
0.4747
0.5455
0.0117
0.003421
a = 10
b = 25
10
Probability that the servers are either idle or on
vacation (
)
=
+
[
15
20
}+
18
} + F(
}]
a = 20
b = 30
16.1302
0.2239
0.6485
0.0778
0.004560
24
26.5239
0.1488
0.7287
0.1224
0.098789
10
14.4438
0.7224
0.3231
0.0007
0.000049
19.6429
0.5106
0.5756
0.0121
0.009878
28.6061
0.3238
0.7076
0.0624
0.023140
45.5809
0.1434
0.7477
0.1109
0.094531
20
(30)
a = 30
b = 50
30
Probability that the server switch over the system
(
)
40
parameters a, b,
,
,
are given in the tables (8.1) to
(8.5).
=[
+k
+
]
(31)
This completes analytic analysis of M/M (a,b)/(2,1)
a=10
a=20
a=30
a=40
a=45
5
10
5.2399
7.5620
9.0993
11.6587
15.2845
15.6054
19.2809
19.9154
21.0005
22.0534
15
20
25
12.0918
16.4365
18.9994
12.7643
15.8790
19.4367
15.9769
17.9896
20.9076
20.1236
21.4732
22.8553
22.4333
23.2970
24.9982
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ISSN 2348-5426
International Journal of Advances in Science and Technology (IJAST)
Vol 2 Issue 2 (June 2014)
Repeated
M/M(a,b)/1
Repeated
M/M(a,b)/(2,1)
Bs
5.5
11.5
17.5
23.5
5.5
11.5
17.5
23.5
9.5
19.5
29.5
39.5
a = 10
b= 30
a = 25
b= 30
a = 40
b= 50
13.665
25.025
42.313
85.086
22.403
35.695
35.085
100.128
26.984
38.860
60.941
120.300
for various values of
, a when b= 50,
Bs
1
1
2
3
0
1
1
3
0
0
1
2
Table 8.1The Performance measures for
= 0.5 and
Single M/M(a,b)/1
5.119
8.129
12.625
18.453
12.044
13.016
16.021
21.516
19.587
21.346
26.559
35.874
Bs
0
0
1
1
1
0
0
1
0
0
0
0
= 0.2 and
= 1Table 8.2
= 18.3:
Comparison
Single M/M(a,b)/(2,1)
Single and delayed
M/M(a,b)/(2,1)
Bs
Bs
11.643
1
5.353
0
23.546
1
8.826
0
42.087
2
13.347
1
83.987
3
18.635
1
15.438
0
12.041
1
30.012
1
12.732
0
51.089
1
14.555
0
97.333
3
17.524
0
20.343
0
19.589
0
30.176
0
20.925
0
53.418
1
24.261
0
119.332
2
29.537
0
of
for M/M (a,b)/1 and M/M(a,b)/(2,1) models
4.999
8.098
13.009
18.323
12.009
12.756
14.112
19.5034
19.6734
20.7903
24.2644
28.8760
0
0
1
1
0
0
0
0
0
0
0
0
The expected total cost per unit time for the operating system M is compared with single and repeated vacation of M/M(a,b)/(2,1)
for various values of a, b when = 0.1 and = 1
Table 8.4
M/M(a,b)/(2,1) Single vacation
M/M(a,b)/(2,1) Repeated
vacation
M
M
5
10
15
20
8
16
24
32
10
20
30
40
Table 8.5
a=10
b=25
a=25
b=40
a= 40
b=50
5.262475
8.98572
16.927956
30.238483
12.347372
16.032642
26.168909
46.064209
19.675701
22.677698
32.856942
55.439793
75.069336
92.09053
118.796974
161.142303
93.351715
109.756927
144.198929
207.200989
114.010056
127.722862
162.452194
233.982773
and M for various values of a, b where
M/M(a,b)/1Repeated vacation
M
5
10
15
20
8
16
24
32
10
20
30
40
a = 10
b= 25
a = 25
b= 40
a = 40
b= 50
51.020
99.760
153.028
227.314
87.517
168.171
256.541
378.552
115.778
217.798
329.634
483.482
182.490
332.725
496.448
723.154
291.698
537.657
806.717
1177.000
376.377
686.235
1026.000
1491.000
5.132
8.404
14.237
23.113
12.289
15.453
23.387
37.406
19.653
22.337
30.752
47.529
= 0.1 and
74.770
90.556
111.643
140.984
93.188
108.081
136.061
181.535
113.943
126.705
156.153
210.265
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5.0123
7.2341
14.2843
28.0001
11.0987
14.1217
24.00081
43.22341
17.47839
21.2345
29.37805
51.42135
71.8076
87.2768
111.6114
148.12263
89.7553
100.0012
138.4390
199.23140
110.3221
120.7685
155.44432
225.64786
=1
M/M(a,b)/(2,1) Repeated
vacation
M
5.132
8.404
14.237
23.113
12.289
15.453
23.387
37.406
19.653
22.337
30.752
47.529
74.770
90.556
111.643
140.984
93.188
108.081
136.061
181.535
113.943
126.705
156.153
210.265
M/M(a,b)/(2,1) single and delayed
vacation
M
M/M(a,b)/2
Nonvacation
M
4.507
4.692
5.338
6.680
12.000
12.047
12.388
13.143
19.500
19.516
19.698
20.417
72.772
79.548
85.666
92.880
92.247
97.443
102.325
108.563
113.449
117.819
121.883
127.038
M/M(a,b)/(2,1) single
delayed vacation
M
3.6723
4.0011
5.2123
5.9021
11.4987
11.8127
11.9921
12.2231
18.1739
18.2645
19.0305
20.4235
72.076
73.2368
82.6394
90.4263
91.0007
97.0012
101.9390
108.2140
112.3721
116.7685
120.6432
121.2786
and
ISSN 2348-5426
International Journal of Advances in Science and Technology (IJAST)
Vol 2 Issue 2 (June 2014)
Choudhury.G and Paul.M(2005),”A two phase
queueing system with Bernoulli Feedback”,
Information and management science,Vol.16,3552
4. Chaudhry.M.L and Easton .G.D (1982),”The
queueing systems Ek/ M(a,b)/1 and its numerical
analysis”, Computer and operations
research,Vol.9,197-205.
5. Doshi.B.T(1986),”Queueing systems with
vacations. A survey”, Queueing systems, Vol.1,
29-66.
6. Ke.J.C(2003),”The optional control of an M/G/1
queueing system with server vacations, startup and
breakdown,”Comput. Indust.Engg.,44: 567 -579.
7. Madan.K.C and AI-Rawwash.M(2005),”On the
Mx/G/1 queue with feedback and optional server
vacations based on a single vacation policy”,
Applied mathematical and computations, Vol 160,
909 -919.
8. Medhi.J.H and Borthakur.A (1972), “On a two
server Markovian queue with a general bulk
service rule”, Cahiers duecentre d’ Etudes de
Recherche operationnelle, Vol.21, 183-189.
9. Mishra.S.S andPandey.N.K (2002), “A Bulk
queueing model M/ M(a,b,c)/2 for non-Identical
servers with vacation”, International journal of
Management and systems, Vol.18, No3, 319-331.
10. Neuts.M.F(1967),”A general class of bulk queues
with Poisson input”, Applied Mathematical and
Statistics,Vol.38, 759 – 770.
11. Palaniammal.S(2004), “A study on Markovian
Queueing models with bulk service and
vacation”,Ph.D, Dissertation, Bharathiar
university, Coimbatore, Tamil nadu, India.
12. Takagi.H(1994),”M/G/1/N queues with server
vacation and exhaustive service”, Journal of
operations research,Vol.42,926-939.
From the table (8.3) we infer that Lq in single and
repeated vacation is less compared to Lq in this single and
delayed vacation only when the difference between the batch
size a and b is less. When the difference between the batch
size a and b is more, the waiting queue in the system is less
in the model with single and delayed vacation compared to
the model with single vacation and repeated vacation this
may be, because of the fact that one server is always retained
in the system.
The table values of (8.5) shows that the number of
batches (of size a and b) waiting in the queue is less by
comparing the other vacation models. It is also seen that Lq
and M are significantly more in M/M(a,b)/1 model
compared to M/M(a,b)/(2,1) queueing model.
9. Conclusion
In this present study, a M/M(a,b)/(2,1) vacation queueing
models with servers vacation depends on the batch sizes and
the state of switch over are considered. In general, analytical
solution of bulk service queueing models are extremely
complicated in the two server’s case. We have made attempt
to study 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. This model is applicable to a variety of real
world stochastic service system. The explicit expressions for
expected queue length may be helpful in setting traffic
management strategies based on performance indices.
References:
1. Afthab Begum . M.I,(1996), “Queueing models
with bulk service and vacation”, Ph.D,
Dissertation, Bharathiar university, Coimbatore,
Tamil nadu, India.
2. Anjana Solanki and Srivastava.P.N(1998),
“Transient state analysis of the queueing system
Ek/ M(a,b)/1/N”, Operations research,
Vol.35,No.4,353-359.
3.
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