www.ijcrt.org © 2020 IJCRT | Volume 8, Issue 4 April 2020 | ISSN: 2320-2882
IJCRT2004061 International Journal of Creative Research Thoughts (IJCRT) www.ijcrt.org 483
OPTIMIZATION OF MULTIPLE SERVER
INTERDEPENDENT QUEUEING MODELS WITH
BULK SERVICE
Dr. Shavej Ali Siddiqui
AssistantProfessor
Faculty of Engineering & Technology
Khwaja Moinuddin Chisti Urdu,Arabi-Farsi University , Lucknow
Sitapur – Hardoi bypass Road , Lucknow – 226013 India
Abstract: In this paper it is devoted to multiple server queuing models having bulk service with interdependence .First the two server
interdependent queuing model with bulk service is developed and then extended to multiple server case. In both the models the explicit
expression are obtained for the steady state system characteristics to analyze their behavior.
Keywords: The system characteristics like, mean queue length, variability of the system size, coefficient of variation are derived and
analyzed in the light of the dependence parameter. These models also include the earlier models as particular cases for specific
values of the parameters.
I. INTRODUCTION
In this paper, two server interdependent queuing model with bulk service is considered for analysis. we assumed that the bulk service is
process is having two service facilities with capacities b1 and b2. A batch of b1 units or the whole queue length whichever is smaller is taken
from the head of the queue for service in the first channel whenever it is free. Similarly, the second channel on becoming free takes b2 (δ b1)
or the whole queue length whichever is less. If both the servers are idle and there is no queue the next unit to arrive always goes to the first
service facility. Here, we assume the two service facilities are independent of each other and the arrival and service processes are
interdependent. These sorts of situations are more common in marshalling yards with two engines, elevator process with two lifts etc. This
model is also extended to multiple server interdependent queuing models with bulk service.
In both the models we first develop the difference-differential equations and solve them through generating function techniques. The system
characteristics are derived and analyzed in the presence of the dependence parameter .These models can be viewed as generalizations of the
Bulk Service queuing models given by Arora (1963)
II. TWO SERVER INTERDEPENDENT QUEUEING MODEL WITH BULK SERVICE
In this paper , the two server queuing model with bulk service having interdependent arrival and service processes is considered, The two
service facilities function independent of each other . We assume the arrivals are poissonian with the mean arrival rate .The marginal
distributions of the service time in the two service facilities are exponential with the mean service rates
1
and
2
. respectively. Then the
conditional process of the number of service completions of the first service facility given that the number of arrivals is of the form,
P(X
11
=n
1
/ X
2
=n
2
; t) = −(
1
−∈
1
) ∑ (
1
)(
∈
1
)
(
1
,
2
)
=0
(
−∈
1
)
2
−
[(
1
−∈
1
)]
1
−
(
1
−)!
…… (1.2.1)
Where
X11 is the number of service completions of the first service facility during time t.
X2 is the number of arrivals during time t.
∈ is the mean dependence rate.
Similarly, the conditional process of the number of service completions of the second service facility given the number of arrivals is of
the form