Int. J. of Mathematical Sciences and Applications,
Vol. 1, No. 3, September 2011
Copyright  Mind Reader Publications
www.journalshub.com
Single Server Interdependent Queueing Model with Fixed Batch Service
Dr. T.S.R.Murthy 1 and D. Siva Rama Krishna 2
1
Dr.T.S.R.Murthy, Professor,
Shri Vishnu Engg. College for Women,
Bhimavaram. West Godavari dist, A.P, India
e-mail: sriram_pavan123@yahoo.co.in
2
D.Siva Rama Krishna, Asst. professor,
Swarnandhra Institute of Engg. & Tech.,
Narsapuram. West Godavari dist, A.P, India
e -mail: srkd999@gmail.com
Abstract
In this paper we extend the single server interdependent queuing models to bulk service queueing
models. Here we consider the costumers/units served a batch of k at a time and if not, the server waits
until such time to start. In this model various system characteristics like average queue length,
probability that the system emptiness and probability that there are n customers at any arbitrary time
are obtained.
Keywords: Queuing system, service process, arrival process, bulk service, interdependence, joint probability, marginal
probabilities.
I
INTRODUCTION
The single server models discussed so far with the interdependent structure assumed that service mechanism is single. Here we
consider a different problem which is a generalization of the earlier queueing models. Here we consider the costumers served a
batch of k at a time and if not, the server waits until such time to start. A typical situation of this model is transportation of men
and material, where the transhipment is the service, men and material are the customers. For this sort of situations in order to have
optimal transshipment, it is appropriate to approximate the situation with the model having the interdependent arrival and service
processes. For developing these interdependent models with bulk service rule, we make use of the dependence structure given by
Rao. K. S(1986).
II
M / M(k)/ 1 INTERDEPENDENT QUEUEING MODEL WITH FIXED BATCH SIZE
In this section, we consider the single server queueing system having the interdependent arrival and processes with bulk service.
Here we consider that the costumers are served a batch of K at a time and if not, the server waits until such times to start. In this
sort of systems the interdependence can be induced by considering the dependence structures having a bivariate Poisson
distribution of the form
P [X 1 = x 1 , X
min .( x 1 , x 2 )
∑
j= 0
2
= x 2 / t ] = e − ( λ + µ −∈ ) t
(∈ t ) j [(λ − ∈ )t ]x − j [(µ − ∈ )t ]x
1
j ! (x 1 − j )! (x
2
2
− j
− j )!
x1,x2 =0, 1, 2, . . . and 0 < λ , µ , ∈< min(λ , µ )
… (1)
P[X1=x1, X2=x2 / t] is the joint probability of x1 arrivals and x2 services during time t.
The marginal distribution of arrival and services are Poisson with parameters λ and µ respectively .Thus inter arrival times and
service times follow negative exponential distributions of the form λ e- λ t and µ e- µt respectively where λ is the mean arrival rate
1065
Dr. T.S.R.Murthy
and D. Siva Rama Krishna
and µ is the mean service rate (Feller 1969). ∈ is the covariance between the number of arrivals and services at time t. This
dependence structure turns out to be independent structure if ∈ = 0 (Teicher1954).
III
POSTULATES OF THE MODEL
The postulates of the model with this dependence structure are
1. The occurrence of the events in non-overlapping time intervals are statistically independent.
2. The probability that no arrivals and no service completions occur in an infinitesimal interval of time ∆t is
1 – [(λ + µ - ∈ ) t] +O (∆t)
3. The probability that no arrival and one service completion occurs in ∆t is
(µ – ∈ ) t + O(∆t)
4. The probability that one arrival and no service completion occurs in ∆t is
(λ – ∈ ) t + O(∆t)
5. The probability that one arrival and no service completion occurs in ∆t is ∈ t + O(∆t).
This postulate is due to the dependence structure between the arrivals and service completions.
6. The probability that the occurrence of an event other than the above events during ∆t is O(∆t)
There is equivalence between the postulates and the process. Further for given values of λ, µ the covariance
r
, where r is the correlation coefficient between arrivals and services. Since ∈ is a function of
treated as dependence parameter. This is the structure given by Rao K.S (1986).
Let Pn(t) be the probability that there are n customers/units in the system at time t.
The difference differential equation of the model are
Pn/ (t ) = ( µ − ∈) Pn+/ k (t ) - (λ − µ + 2 ∈) Pn/ (t ) + (λ − ∈) Pn−/ 1 (t )
P (t ) = ( µ − ∈) P (t ) - (λ − ∈) P (t ) + (λ − ∈) P (t )
/
n+ k
/
n
/
n
/
n−1
for
n≥k
for 1 ≤ n p k
… (2)
P0/ (t ) = - (λ − ∈) P0/ (t ) + ( µ − ∈) Pk/ (t ) for n = 0
…(3)
Assuming that the system is reached study state, the transition equations of model are
( µ − ∈) Pn + k - (λ + µ − 2 ∈) Pn + (λ − ∈) Pn−1 =0
n≥k
( µ − ∈) Pn+ k - (λ − ∈) Pn + (λ − ∈) Pn−1 =0 for 1 ≤ n p k
for
…(4)
- (λ − ∈) P0 + ( µ − ∈) Pk =0
for n=0 . . . (5)
Using the non homogeneous linear difference equation technique, we obtain
p
n

1 −
 P0
1

= 
 P  λ
 0  µ
r n +1
1 ≤ n p k
− r
− ∈  n -k
 r
n ≥ k
− ∈ 
… (6)
Where r is the root which lie in the interval (0,1) of the characteristic equation
{(µ − ∈) D
k +1
- (λ + µ − 2 ∈) D +
(λ − ∈)}Pn = 0
for n ≥ k … (7)
Where D is the operator.
1066
∈=
r, throughout this paper ∈
Single Server Interdependent Queueing Model
Using the bounded conditions Pi ≥ 0 and
We obtain that
P0 =
∑P
n
=1
1− r
… (8)
k
The Probability that there are n customers in the system at any arbitrary time is
pn
1
n +1
)
1 ≤ n p k
 k (1 - r

= 
  1 − r   λ − ∈  r n - k
n ≥ k
  k   µ − ∈ 
… (9)
Where n is as given in the equation (6)
Now the probability generating function of the model is
k
n
P ( Z ) = (1 − z ) ∑ p n z
Where
ρ
ρ z k +1 − ( p + 1) z k + 1
= λ−∈
µ−∈
Using Rouche’s theorem if we denote the root of the characteristic equation which lies out side the unit circle is Z0 then
P(Z ) =
( z 0 − 1) ∑ z n
Where
z0 =
1
… (10)
r
( z0 − z)k
From equation (9), we observe that there is recursive relation for the probability mass function
pn = pk r n−k
4
n ≥ k where r is given above equation (7)
MEASURES AND EFFECTIVENESS
P0 =
The probability of the system is empty if
1− r
k
Where r is given in the equation (7)
For fixed values of λ and µ and for various values of k and ∈ we computed the values of r and are given in table (1.1) and the
values of r for fixed k, ∈ and for varying values of λ, µ are given in table (1.3). For various values of ∈ and k for given value of λ
and µ we computed P0 values and are given in the table (1.2) the values of P0 for fixed k, ∈ for varying λ, µ are given in the table
(1.4)
Table 1.1 The values of r
where λ = 3 and µ = 4
∈
K
0
0.1
0.2
0.3
1067
0.4
0.5
0.6
0.7
0.8
Dr. T.S.R.Murthy
and D. Siva Rama Krishna
2
0.5
0.4968
0.4934
0.4898
0.486
0.482
0.4777
0.4731
0.4682
3
0.4525
0.45
0.4473
0.4444
0.4414
0.4382
0.4347
0.4311
0.4271
4
0.4378
0.4354
0.433
0.4304
0.4277
0.4247
0.4216
0.4183
0.4147
5
0.4323
0.4301
0.4278
0.4253
0.4227
0.4199
0.4169
0.4137
0.4102
6
0.4301
0.428
0.4257
0.4233
0.4207
0.418
0.415
0.4119
0.4085
7
0.4292
0.4271
0.4249
0.4225
0.4199
0.4172
0.4143
0.4112
8
0.4289
0.4267
0.4245
0.4221
0.4196
0.4169
0.414
0.4109
9
0.4287
0.4266
0.4244
0.422
0.4195
0.4168
0.4139
0.4108
0.5
0.259
0.187267
0.143825
0.11602
0.097
0.083257
0.072888
0.0648
0.6
0.26115
0.188433
0.1446
0.11662
0.0975
0.083671
0.07325
0.065122
where λ = 3 and µ = 4 where
Table 1.2 The values of P0
∈
K
2
3
4
5
6
7
8
9
0
0.25
0.1825
0.14055
0.11354
0.094983
0.081543
0.071388
0.063478
0.1
0.2516
0.183333
0.14115
0.11398
0.095333
0.081843
0.071663
0.063711
Table 1.3: The values of r
λ
µ
0.2
0.2533
0.184233
0.14175
0.11444
0.095717
0.082157
0.071938
0.063956
0.3
0.2551
0.1852
0.1424
0.11494
0.096117
0.0825
0.072238
0.064222
0.4
0.257
0.1862
0.143075
0.11546
0.09655
0.082871
0.07255
0.0645
0
0.263
0.1896
0.1454
0.117
0.0980
0.0841
0.0736
0.0654
where k = 2 and ∈ = 0.5
1
2
3
4
5
6
7
8
9
10
0.0501
0.1387
0.2164
0.2864
0.3507
0.4105
0.4665
0.5195
0.5699
11
0.0455
0.1268
0.1986
0.2638
0.3238
0.3797
0.4322
0.482
0.5293
12
0.0417
0.1168
0.1837
0.2445
0.3008
0.3534
0.4029
0.4498
0.4946
13
0.0385
0.1083
0.1708
0.228
0.281
0.3307
0.3775
0.422
0.4644
14
0.0358
0.1009
0.1597
0.2136
0.2638
0.3108
0.3553
0.3975
0.4379
1068
Single Server Interdependent Queueing Model
15
0.0334
Table 1.4:
λ
µ
0.0945
0.1499
The values of P0
0.201
0.2486
0.2933
0.3356
0.3759
0.4144
where k = 2 and ∈ = 0.5
1
2
3
4
5
6
7
8
10
0.47495
0.43065
0.3918
0.3568
0.32465
0.29475
0.26675
0.24025
0
11
0.47725
0.4366
0.4007
0.3681
0.3381
0.31015
0.2839
0.259
0
12
0.47915
0.4416
0.40815
0.37775
0.3496
0.3233
0.29855
0.2751
13
0.48075
0.44585
0.4146
0.386
0.3595
0.33465
0.31125
0.289
14
0.4821
0.44955
0.42015
0.3932
0.3681
0.3446
0.32235
0.30125
15
0.4833
0.45275
0.42505
0.3995
0.3757
0.35335
0.3322
0.31205
Table 1.5 values of Pn when 1 ≤ n p k
0
where k=9, λ = 3 and µ = 4
∈
n
0
0.1
0.2
0.3
0.4
0.5
0.6
0.
1
0.090691
0.09089
0.091098
0.091324
0.091558
0.091809
0.092076
0.0923
2
0.102357
0.102485
0.102618
0.102761
0.102908
0.103066
0.103233
0.10340
3
0.107358
0.107431
0.107506
0.107587
0.10767
0.107758
0.10785
0.10794
4
0.109502
0.109541
0.109581
0.109624
0.109668
0.109713
0.109761
0.10981
5
0.110421
0.110441
0.110462
0.110484
0.110506
0.110529
0.110552
0.11057
6
0.110815
0.110825
0.110836
0.110846
0.110857
0.110868
0.11088
0.11089
7
0.110984
0.110989
0.110994
0.110999
0.111005
0.11101
0.111015
0.11102
8
0.111057
0.111059
0.111061
0.111064
0.111066
0.111069
0.111072
0.11107
Table 1.6
values of Pn where n ≥ k
0
0.1
where
k=9, λ = 3 and µ = 4
0.2
0.3
1069
0.4
0.5
0.6
Dr. T.S.R.Murthy
and D. Siva Rama Krishna
∈
n
10
0.020409693
0.02021
0.020003
0.019777
0.019542
0.019292
0.019026
11
0.008749635
12
0.003750969
13
0.008622
0.00849
0.008346
0.008198
0.008041
0.007875
0.00
0.003678
0.003604
0.003522
0.003439
0.003351
0.003259
0.0031
0.00160804
0.001569
0.00153
0.001486
0.001443
0.001397
0.001349
0.0012
14
0.000689367
0.000669
0.000649
0.000627
0.000605
0.000582
0.000558
0.0005
15
0.000295532
0.000286
0.000276
0.000265
0.000254
0.000243
0.000231
0.0002
16
0.000126694
0.000122
0.000117
0.000112
0.000107
0.000101
9.57E-05
9.01E
17
5.43139E-05
5.2E-05
4.97E-05
4.71E-05
4.47E-05
4.22E-05
3.96E-05
3.7E
18
2.32844E-05
2.22E-05
2.11E-05
1.99E-05
1.87E-05
1.76E-05
1.64E-05
1.52E
Table 1.7 values of Pn where 1 ≤ n p k
λ
n
1
2
3
4
5
6
7
8
1
0.109375
0.110894
0.111084
0.111108
0.111111
0.111111
0.111111
0.111111
Table 1.8
λ
n
2
0.101111
0.108111
0.110211
0.110841
0.11103
0.111087
0.111104
0.111109
where k=9, ∈= 0.5 and µ = 4
3
0.091809
0.103066
0.107758
0.109713
0.110529
0.110868
0.11101
0.111069
values of Pn where n ≥ k
0.0187
4
0.083278
0.097181
0.104139
0.107621
0.109365
0.110237
0.110674
0.110892
5
0.07578
0.091188
0.099876
0.104776
0.107539
0.109097
0.109975
0.110471
6
0.069209
0.085379
0.095309
0.101407
0.105152
0.107452
0.108864
0.109731
7
0.063427
0.079873
0.090647
0.097705
0.102329
0.105358
0.107342
0.108642
8
0.058272
0.074674
0.085984
0.093783
0.099162
0.102871
0.105429
0.107192
where k=9, ∈= 0.5 and µ = 4
1
2
3
4
5
6
7
10
0.001736111
0.01
0.019292
0.027778
0.035131
0.041378
0.046623
0.0
11
0.000217014
0.003
0.008041
0.013903
0.01981
0.02541
0.030543
0.0
12
2.71267E-05
0.0009
0.003351
0.006958
0.011171
0.015604
0.020009
0.0
13
3.39084E-06
0.00027
0.001397
0.003483
0.006299
0.009583
0.013108
0.0
14
4.23855E-07
0.000081
0.000582
0.001743
0.003552
0.005885
0.008587
0.0
15
5.29819E-08
2.43E-05
0.000243
0.000872
0.002003
0.003614
0.005625
0.0
16
6.62274E-09
7.29E-06
0.000101
0.000437
0.00113
0.002219
0.003685
0.0
17
8.27842E-10
2.19E-06
4.22E-05
0.000219
0.000637
0.001363
0.002414
0.
1070
Single Server Interdependent Queueing Model
18
Table 1.9
µ
n
1.0348E-10
6.56E-07
values of Pn where 1 ≤ n p k
1.76E-05
0.000109
0.000359
0.000837
0.001581
0.0
10
11
where k=9, ∈= 0.5 and λ = 3
4
5
6
7
8
9
1
0.091809
0.096934
0.10026
0.102536
0.104167
0.105371
0.10629
0.107002
2
0.103066
0.106047
0.10772
0.108729
0.109375
0.109806
0.110107
0.110321
3
0.107758
0.109302
0.110051
0.110449
0.110677
0.110815
0.110902
0.110959
4
0.109713
0.110465
0.11078
0.110927
0.111003
0.111044
0.111068
0.111082
5
0.110529
0.11088
0.111008
0.11106
0.111084
0.111096
0.111102
0.111105
6
0.110868
0.111029
0.111079
0.111097
0.111104
0.111108
0.111109
0.11111
7
0.11101
0.111082
0.111101
0.111107
0.111109
0.11111
0.111111
0.111111
8
0.111069
0.111101
0.111108
0.11111
0.111111
0.111111
0.111111
0.111111
Table 1.10
µ
n
Values of Pn where n ≥ k
where k=9, ∈= 0.5 and λ = 3
4
5
6
7
8
9
10
10
0.02700864
0.032801
0.037512
0.041399
0.044643
0.047393
0.049735
0.05
11
0.011257201
0.011717
0.011723
0.011501
0.011161
0.010773
0.01036
0.00
12
0.004692001
0.004185
0.003663
0.003195
0.00279
0.002449
0.002158
0.00
13
0.001955626
0.001495
0.001145
0.000888
0.000698
0.000557
0.00045
0.00
14
0.000815105
0.000534
0.000358
0.000247
0.000174
0.000127
9.36E-05
7.08
15
0.000339736
0.000191
0.000112
6.85E-05
4.36E-05
2.88E-05
1.95E-05
1.36
16
0.000141602
6.81E-05
3.49E-05
1.9E-05
1.09E-05
6.54E-06
4.06E-06
2.62
17
5.90197E-05
2.43E-05
1.09E-05
5.29E-06
2.72E-06
1.49E-06
8.46E-07
5.03
1071
Dr. T.S.R.Murthy
18
2.45994E-05
Table 1.11
K
n
8.69E-06
and D. Siva Rama Krishna
3.41E-06
values of Pn where 1 ≤ n p k ,
1.47E-06
6.81E-07
3.38E-07
for fixed λ = 3, µ = 4 and ∈= 0.5
5
6
7
8
9
10
1
0.164736798
0.137546
0.117992
0.103274
0.091809
0.082636
2
0.185192981
0.154494
0.132483
0.115943
0.103066
0.092764
3
0.193782533
0.161579
0.138529
0.121224
0.107758
0.096985
4
0.197389286
0.16454
0.141052
0.123426
0.109713
0.098744
Table 1.12
K
n
Values of Pn where n ≥ k ,
1.76E-07
for fixed λ = 3, µ = 4 and ∈= 0.5
5
6
7
8
9
10
11
0.000454234
0.000884
0.001802
0.003772
0.008041
0.017362
12
0.000190733
0.00037
0.000752
0.001573
0.003351
0.007235
13
8.00888E-05
0.000154
0.000314
0.000656
0.001397
0.003015
14
3.36293E-05
6.46E-05
0.000131
0.000273
0.000582
0.001256
From table (1.5) and equation (9) we observe that for fixed values of k, µ, λ and n the values of Pn increases as the
dependence parameter ∈ increases. For fixed values of k, µ, λ and ∈ the values of Pn increases as n increases and the values of
p0 decreases for n ≥ k which is shown table (1.6) for the same values. From table (1.7) the values of pn for 1 ≤ n p k ,for
fixed values of k, µ, ∈ and n decreases as λ increases and for fixed values of k, µ, λ , ∈ , the values of Pn increases as n increases.
1072
9.68
Single Server Interdependent Queueing Model
From table (1.8) the values of Pn for n ≥ k , for fixed values of k, µ, ∈ and n increases as λ increases and for fixed values of k, µ,
λ and ∈ the values of Pn for n ≥ k decreases as n increases. From table (1.9) the values of Pn for 1 ≤ n p k , for fixed values
of k, λ, ∈ and n increases as µ increases and for fixed values of k, µ, λ and ∈ increases as n increases. From table (1.10) the
values of Pn for
n ≥ k for fixed values of k, λ and ∈ and n decreases as µ increases and for fixed values of k, µ, λ and ∈ the
values of Pn decreases as n increases. From table (1.11) the values of Pn for
1 ≤ n p k , for fixed values of λ, µ , ∈ and k
increases as n increases and for fixed values of µ, λ , ∈ and n decreases as k increases. From table (1.12) the values of Pn for
n ≥ k for fixed values of λ, µ, ∈ and k decreases as n increases and for fixed values of µ, λ, ∈ and k increases as k increases.
The probability that the server is busy is obtained
=
1 −
k −1
∑
i= 0
i+1
 k
 1 − r

 
1
−
r
1
−
r





2
= 1 −   k  −  k r

 (1 − r )3
  1 − r 


 1 − r k −1

(
 … (11)

)
From tables (1.2), (1.4) and equation (8) we observe that for fixed values of λ, µ and ∈ the values of P0 decreases as k increases.
As the dependence parameter ∈ increases the value of P0 increases for fixed values of λ, µ and k. The value of P0 decreases for
1073
Dr. T.S.R.Murthy
and D. Siva Rama Krishna
fixed values of µ, k and ∈ as λ increases. As µ increases the value of P0 increases for fixed values of µ, k and the dependence
parameter ∈. If the mean dependence rate is zero then the value of P0 is same as in the M / M(k) / 1 model.
∞
Ls = ∑ np n =
The average numbers of customer in the system are obtained
n =0
k −1
r
+
2
1− r
..
. (12)
The average number of customers in the queue
Lq ==
k −1
r
k −1+ r
+
−
2
1− r
k
. . . (13)
Where r is given in the equation (7)
The values of Ls and Lq are computed and are given in the tables (1.13) and (1.15) for given values of λ, µ and for
various values of ∈ and k respectively. The values of Ls and Lq are fixed values of ∈, k and for varying values of µ and λ are also
given in tables (1.14) and (1.16)
Table 1.13
Values of
Ls
λ = 3 and µ = 4
∈
K
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
2
1.5
1.487281
1.473944
1.460016
1.445525
1.430502
1.414608
1.397893
3
1.826484
1.818182
1.8093
1.799856
1.79019
1.779993
1.768972
1.757778
4
2.278726
2.271165
2.263668
2.255618
2.247335
2.238224
2.228907
2.219099
5
2.761494
2.754694
2.747641
2.740038
2.732202
2.723841
2.714972
2.705611
6
3.254694
3.248252
3.24125
3.234004
3.226221
3.218213
3.209402
3.200391
7
3.751927
3.745505
3.738828
3.731602
3.723841
3.715854
3.707359
3.69837
8
4.251007
4.244287
4.237619
4.230403
4.22295
4.214972
4.206485
4.197505
9
4.750394
4.743983
4.737318
4.730104
4.722653
4.714678
4.706193
4.697217
Table 1.14 The values of
λ
µ
Ls
k = 2 and ∈ = 0.5
1
2
3
4
5
6
7
8
10
0.552742
0.661036
0.776161
0.901345
1.04012
1.196353
1.374414
1.581165
11
0.547669
0.645213
0.747816
0.858327
0.978852
1.112123
1.261184
1.430502
1074
Single Server Interdependent Queueing Model
12
0.543515
0.632246
0.72504
0.823627
0.930206
1.046551
1.174761
1.317521
13
0.540042
0.621453
0.705982
0.795337
0.890821
0.994098
1.106426
1.230104
14
0.537129
0.612223
0.690051
0.771617
0.858327
0.950958
1.051109
1.159751
15
0.534554
0.604362
0.676332
0.751564
0.830849
0.915028
1.005117
1.102307
Table 1.15
λ = 3 and µ = 4
Values of L q
∈
K
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
2
0.75
0.738881
0.727244
0.715116
0.702525
0.689502
0.675758
0.661343
3
1.008984
1.001515
0.993533
0.985056
0.97639
0.96726
0.957406
0.947412
4
1.419276
1.412315
1.405418
1.398018
1.39041
1.382049
1.373507
1.364524
5
1.875034
1.868674
1.862081
1.854978
1.847662
1.839861
1.831592
1.822871
6
2.349677
2.343585
2.336967
2.33012
2.322771
2.315213
2.306902
2.298408
7
2.83347
2.827348
2.820985
2.814102
2.806712
2.799112
2.79103
2.782484
8
3.322394
3.31595
3.309557
3.302641
3.2955
3.287859
3.279735
3.271142
9
3.813872
3.807694
3.801273
3.794326
3.787153
3.779478
3.771316
3.762683
Table 1.16 Values of L q
λ
µ
k = 2 and ∈ = 0.5
1
2
3
4
5
6
7
10
0.027692
0.091686
0.167961
0.258145
0.36477
0.491103
0.641164
0.82141
11
0.024919
0.081813
0.148516
0.226427
0.316952
0.422273
0.545084
0.68950
12
0.022665
0.073846
0.13319
0.201377
0.279806
0.369851
0.473311
0.59262
13
0.020792
0.067303
0.120582
0.181337
0.250321
0.328748
0.417676
0.51910
14
0.019229
0.061773
0.110201
0.164817
0.226427
0.295558
0.373459
0.46100
15
0.017854
0.057112
0.101382
0.151064
0.206549
0.268378
0.337317
0.41435
1075
Dr. T.S.R.Murthy
and D. Siva Rama Krishna
From equations (12), (13) and from the corresponding tables we observe that as ∈ increases the values of
decreasing and also as k increases the value of
Ls and Lq are increasing for fixed values of other parameters.
As the arrival rate increases the values of
values of
Ls and Lq are
Ls and Lq are increasing for fixed values of µ, k and ∈. As µ increases the
Ls and Lq are decreasing for fixed values of λ, k and ∈. When the dependence parameter ∈=0 then the average
queue length is same as that of M/M(k)/1 model. When k=1 this is same as M/M/1 interdependence model.
CONCLUSION
In this paper, we extend the single server interdependent queueing models to bulk service queueing models with fixed
batch service. Where the arrival and service patterns are interdependent. These models have wider applicability in
transportation, inventory control, machine interference problems, neurophysiological systems etc. For efficient design and to
predict the system performance measures, the system behavior is analyzed by using the system characteristics like average
number of costumers in the system and queue, probability of the system emptiness and probability that there are n customers in
the system at any time t. In this paper, it is observed that the positive dependence between the arrival and the service
completions can reduce the mean queue length. This is a more useful result in developing the optimal operating policies.
REFERENCES
[1]
ATTAHIRU SULE ALFA (1982), TIME
INHOMOGENEOUS Bulk Server Queue in Discrete Time – A
Transportation
Type Problem, Opern. Res., 30.
[2] Aurora, k. L.(1964), Two Server Bulk Service Queueing Process, Opern.
Res. ,
12.
[3] BAITEN, N.T.J.(1954), On Queueing Processes with Bulk Service, J.Roy.Soc., B-16.
[4] BHAT,U.N.(1964), On Single Server Queueing Process with Binomial Input, Opern. Res.,12
[5]
BORST, S.C. et.al(1993), An M/G/1 Queue with Customer Collection, Stochastic Models., 9.
[6] CONOLLY (1968), The Waiting Time Process for a Certain Correlated Queue., Opern. Res., 16.
[7]
DOWNTON,F.(1955), Waiting Time in Bulk Service
Queues, J.Roy.Stat.Soc.,17
[8] ERNEST, M.SCHEVER (1973), Maximum and Minimum Service times for batches of jobs with an Unknown Number
of
Service, Opern. Res., 21.
[9] FELLER,W.(1973), An Introduction to Probability theory and Its Applications, Vol.II Wiley, New York.
[10] FOSTER, F.G.& NYUNT, K.M (1961), Queues with Opern. Res.12
[11] GHARE, P.M.(1968), Multichannel Queueing System with Bulk Service, Opern. Res., 16.
[12] GOYAL.J.K.(1967), Queues with Hyper Poisson Arrivals and Bulk Exponential Service, Metrika, 11
[13] GROSS and HARRIS (1974), Fundamentals of
Queueing Theory, john Wiley & Sons, New York.
[14] JAISWAL,N.K.(19610) Bulk Service Queueing Problem Opern. Res., 8
[15] KENDALL,D.G.(1951), Some Problem in the theory of Queues, J. Roy. Stat Soc.-(B), 13.
[16] LINDLEY (1952). The Theory of Queues with Single Server, Proc. Camb. Phil. Soc., 48
[17] MITCHELL and PAULSON (1979). M/M/1 Queues with Interdependent Arrival and Service Processes, Nov Res.
Logist. Quart., 26
[18] MURTHY,T.S.R.(1993), Some Waiting Line Models with Bulk Service, Ph.D. Thesis Andhra University.
[19] NEUTS , M.F. (1965), The Busy Period of Queue with Batch Service, Operan. Res., 13
[20] RAO,K.S.(1986), Queues with Input – Output Dependence VIII ISPS Annual Conference, held at Kolhapur.
[21] Dr T S R Murthy and D S R Krishna (2011), Interdependent queueing model with varying bulk service, RTAMST,
Githam University, Visakhapatnam.
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Single Server Interdependent Queueing Model
[22] Dr T S R Murthy and D S R Krishna (2011), Interdependent queueing model with fixed size service, ICACCN-11, 23 June, Chandigarh.
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