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Advances and Applications in Mathematical Sciences Volume 20, Issue 6, April 2021, Pages 975-1001 © 2021 Mili Publications
2010 Mathematics Subject Classification: 05C78.
Keywords: breakdown, delay time, optional vacation.
Received December 10, 2019; Accepted May 15 2020
ANALYSIS OF A MULTI TYPE SERVICE OF A NON-
MARKOVIAN QUEUE WITH BREAKDOWN, DELAY TIME
AND OPTIONAL VACATION
P. MANOHARAN and K. SANKARA SASI
Department of Mathematics
Annamalai University
Annamalainagar-608 002, India
E-mail: [email protected]
[email protected]
Abstract
We consider an 1GM queuing system with k-types of service, random breakdowns, delay
times for repairs to start and a second optional vacation. All arriving customers may choose
either of type j services with probability jp where
k
jj kjp
1.,,3,2,1,1 When the
system becomes empty, the server goes for regular vacation and at the end of the first vacation
the server may take a second optional vacation with .1 pro bability , otherwise he
remains in the system with probability. The system may breakdown at random, its repairs do
not start immediately and there is a delay time. The delay times and the repair times follow a
general distribution. Using supplementary variable technique, we derive the probability
generating function for the number of customers in the system, the average number of
customers in the system and the average waiting time of customers in the system. Particular
case is deduced to check the validity of the present model with already existing models.
Numerical examples are also provided.
1. Introduction
In many examples such as production system, bank services, computer
and communication networks, besides feedback the system have vacation.
Vacation queues with different vacation policies including Bernoulli
schedules, assuming a single vacation policy or multiple vacation policy have
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P. MANOHARAN and K. SANKARA SASI
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976
been studied by many researchers. Levy and Yechiali [12], Fuhrman [9],
Doshi [7] and [8], Keilson and Servi [11], Baba [1], Cramer [6], Borthakur and
Chaudhury [3], Madan [13], [14] and [15], Choi and Park [5], Takagi [18] and
[19], Rosenberg and Yechiali [17] ,Chaudhury [4], Badamchi Zadeh and
Shankar [2] and many others have studied vacation queues with different P.
Manoharan and K. Sankara Sasi vacation policies. Madan and Chaudhury
[16] have studied a single server queue with two phase of heterogeneous
service under Bernoulli schedule and a general vacation time. In this system,
without feedback, the server after completing the service can take vacation
with probability or remain in the system with probability 1 Madan
and Anabosi [15] have studied a single server queue with optional server
vacations based on Bernoulli schedules and a single vacation policy. In this
system, without feedback, the server provides two types of heterogeneous
exponential service and a customer may choose either type of service.
Moreover, the server after completing the service can take vacation with
probability or remain in the system with probability .1
In this paper, we consider an 1GM queuing system with k-types of
service, random breakdowns, delay times for repairs to start and a second
optional vacation. Using supplementary variable technique, we derive the
probability generating function for the number of customers in the system,
the average number of customers in the system and the average waiting time
of customers in the system. The paper is organaized as follows. In section two
the model is described. In section three the distribution of the system is
obtained. In section four the performance measures are calculated. In section
five a particular case is discussed. In section six numerical illustrations are
presented.
2. Model Description
The arrival follows Poisson distribution with mean arrival rate .0
The server provides k-types of service to all arriving customers. Customer
may choose either of type j services with probability jp where
k
j jp1
.1
The service times follows a general distribution, with distribution function
xB j and density function xb j for .,,2,1 kj Further it is assumed
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that dxxj is the conditional probability of completion of the jth service
given that the elapsed service time is x, so that
xB
xbdxx
j
j
j
1
(1)
and therefore
.,,2,1,0
kjexxb
x
j dtt
jj
We assume that the services are mutually independent of each other. Let
kjcBEsBc
jj ,,2,1,1,, denote the Laplace-Stieltjes Transform
(LST) and finite moments of service times respectively. Thus the total time
required by the server to complete a service cycle which may be called as
modified service period is given by
.yprobabilitwith
yprobabilitwith
yprobabilitwith
22
11
kk pB
pB
pB
B
The system may breaks down at random, and the breakdowns are
assumed to occur according to a Poisson stream with mean breakdown rate
.0 Further we assume that once the system breaks down, the customer
whose service is interrupted, comes back to the head of the queue. Once the
system breaks down, its repairs do not start immediately and there is a delay
time. The delay times follow a general (arbitrary) distribution with
distribution function xS and density function .xs Let xdxv be the
conditional probability of a completion of the delay process given that the
delay time is x, so that
xS
xsdxxv
1 (2)
and hence
.0
xdttv
exvxs
Further, the repair times follow a general distribution with distribution
function xG and density function .xg Further it is assumed that dxx
is the conditional probability of the completion of the repair process given
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978
that the elapsed repair time is x, so that
xG
xgdxx
1 (3)
and hence
.0
xdtt
exxg
Whenever the system becomes empty, the server goes for a first phase of
regular vacation (FRV) of random length .1V Let xV1 and xv1
respectively denote the distribution function and density function of the first
vacation time. At the end of FRV, the server may take a second optional
vacation (SOV) with probability , otherwise he remains in the system with
probability 1 until a new customer arrives. Let xV 2 and xv 2
respectively denote the distribution function and density function for the SOV
time. Further it is assumed that dxxv i is the conditional probability of the
completion of the ith vacation given that the elapsed vacation time is x, so that
xV
xvdxxv
i
ii
1 (4)
and hence
.2,1;0
iexvxv
x
i dttv
ii
It is also assumed that the vacation times 1V and 2V are mutually
independent of each other having LSTs sV i and finite moments,
.2,1,1, ikVEk
i Thus the total time required to complete the
vacation cycle, which may be called as modified vacation period is given by
.1yprobabilitwith
yprobabilitwith
1
21
V
VV
V
3. System size Distribution
We set up the steady state equations for the stationary queue size
distribution by treating elapsed service time, delay time, repair time, FRV
time and SOV time as supplementary variables. Using these equations, we
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derive the probability generating functions, assuming that the system is in
steady state condition. Let tN be the system size (including one being
served, if any), tB
j
0 be the elapsed service time at t for type tSj
0, be
the elapsed delay time at tRt
0, be the elapsed repair time at
tVt
0
1, be
the elapsed vacation time at t for the FRV,
tV0
2 be the elapsed vacation
time at t for the SOV. For further development of this model, introduce the
random variable tY as follows.
t
t
tj
t
t
tY
time atrepair under is system the if4
time atrepair for delayunder is system the if3
time at service typeproviding busy isserver the if2
time at SOVon isserver the if1
time at FRV on isserver the if0
The supplementary variables
tSkjtBtVtVj
0002
01 ,,,2,1;,, and
tR
0 are introduced in order to obtain a bivariate Markov process
0;, tttN where
4if
3if
2if
1if
0if
0
0
0
02
0
1
tYtR
tYtS
tYtB
tYtV
tYtV
tj
and we define the limiting probability as follows.
0,0,,,Prlim0
1
0
1,1
xndxxtVxtVtntNdxxQt
n
0,0,,,Prlim0
2
0
2,2
xndxxtVxtVtntNdxxQt
n
0,0,,,Prlim00
,
xndxxtBxtBtntNdxxPjj
tnj
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0,0,,,Prlim00
xndxxtSxtStntNdxxDt
n
.0,0,,,Prlim00
xndxxtRxtRtntNdxxRt
n
Further it is assumed that ,1,00;1,00 SSBB jj
1,00 RR and are continuous at ,0x where 1,00 ii VV
are distribution functions for kj ,,2,1 and 2,1i so that,
xG
xdGdxx
xS
xdSdxxv
xB
xdBdxx
j
j
j
11
;1
and
.
1 xV
xdVdxxv
i
ii
The differential-difference equations governing the system are
kjnxPxPxxPdx
dnjnjjnj ,,2,1,1,1,,, (5)
kjxPxxPdx
dnjjj ,,2,1,0,0, (6)
1,1 nxDxDxvxDdx
dnnn (7)
0,00 xxDdx
d (8)
1,1 nxRxRxxRdx
dnnn (9)
,000 xRxxRdx
d (10)
1,1,1,11,1 nxQxQxvxQdx
dnnn (11)
00,110,1 xQxvxQdx
d (12)
1,1,2,22,2 nxQxQxvxQdx
dnnn (13)
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,00,220,2 xQxvxQdx
d (14)
k
m
mm dxxxRdxxxPQ
10
00
0,0,1 1
0 0
20,210,1 dxxvxQdxxvxQ (15)
where
.0
0,10,1
dxxQQ
The boundary conditions are
0,10,1 0 QQ (16)
1,00,1 nQ n (17)
0
1,1,2 0,0 ndxxvxQQ nn (18)
k
m
jmmjj dxxxRpdxxxPpP
10 0
11,0, 0
0
11,11 dxxvxQpp jj
0
11,2 ,,2,1 kjdxxvxQp j (19)
k
m
njmnmjnj dxxxRpdxxxPpP
10 0
11,, 0
0
11,11 dxxvxQp nj
0
11,2 ,,2,1 kjdxxvxQp nj (20)
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k
m
nmn ndxxPD
10
1, 1,0 (21)
000 D (22)
0
1,0 ndxxxDR nn (23)
000 R (24)
and the normalizing condition is
10
1 10
,
n
n
n m
nm dxxRdxxP
0
2
10
,
10
.1
n i
ni
n
n dxxQdxxD (25)
For ,2,1;,,2,1,1;0 ikjzx we define the following
Probability Generating Functions
00
,
0
, ,;,0;,
n
jjnjn
n
jnjn
j dxzxPzPxPzzPxPzzxP
0
00
,;0,0;, dxzxDzDDzzDxDzzxD
n
nn
n
nn
0
00
,;,0;, dxzxRzRxRzzRxRzzxR
n
nn
n
nn
0
0
,
0
, .,;0,0;, dxzxQzQQzzQxQzzxQ i
n
inin
n
inin
i
Multiplying equation (5) by nz and summing from 1n to ∞ and adding
the resultant with equation (6), we get
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983
zxzPzxPxzxPdx
djjjj ,,,
.
,
,
xzzxP
zxPdx
d
jj
j
Integrating the above equation with respect to x between 0 and x, we get
.,0
,0
x
j dttxz
j
je
zP
zxP
From equation (1)
.,,2,1,
1kj
tB
tbdtt
j
j
j
Integrating the above equation with respect to x between 0 and x, we get
.,,2,1,10
kjxBe j
dttx
j
(26)
Using equation (26) in (25), we get
.,,2,1;1,0, kjexBzPzxPxz
jjj
(27)
Multiplying equation (7) by nz and summing from 1n to ∞ and adding the
resultant with equation (8), we get
zxzDzxDxvzxDdx
d,,,
xvz
zxD
zxDdx
d
,
,
Integrating the above equation with respect to x between 0 and x, we get
.,0
, 0
xdttvxx
ezD
zxD (28)
From equation (2)
.
1 ts
tsdttv
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Integrating the above equation with respect to x between 0 and x, we get
.10
xSe
xdttv
(29)
Using equation (29) in (28), we get
.1,0,xz
exSzDzxD
(30)
Multiplying equation (9) by nz and summing from 1n to ∞ and adding the
resultant with equation (10), we get
zxRzzxRxzxRdx
d,,,
.
,
,
xzzxR
zxRdx
d
Integrating the above equation with respect to x between 0 and x, we get
.,0
, 0
xdttxx
ezR
zxR (31)
From equation (3)
.
1 tG
tgdtt
Integrating the above equation with respect to x between 0 and x, we get
.10
xGe
xdtt
(32)
Using equation (32) in (31), we get
.1,0,xz
exGzRzxR
(33)
Multiplying equation (11) by nz and summing from 1n to ∞ and adding
the resultant with equation (12), we get
.1,0, 121xx
exVzQzxQ
(34)
Multiplying equation (13) by nz and summing from 1n to ∞ and adding
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the resultant with equation (14), we get
.1,0, 222xz
exVzQzxQ
(35)
Multiplying equation (17) by nz and summing from 1n to ∞ and using
equation (16), we get
.,0 0,11 QzQ (36)
Multiplying equation (18) by nz and summing from 0n to ∞, we get
.,0,0 112 zVzQzQ (37)
Multiplying equation (20) by 1nz and summing from 1n to ∞ and adding
with z times equation (19), we get
k
m
jmmjj dxxzxRpdxxzxPpzzP
100
,,,0
.,,10
0,1220
11
QpdxxvzxQdxxvzxQp jj (38)
Multiplying equation (21) by nz and summing from 1n to ∞, we get
.,0
1
k
m
m zPzzD (39)
Multiplying equation (23) by nz and summing from 1n to ∞, we get
0
,,0 dxxvzxDzR (40)
Multiplying equation (27) by xj and integrating over 0 to ∞, we get
.,,3,2,1,,0,0
kjzBzPdxxzxP jjjj
(41)
Multiplying equation (30) by xv and integrating over 0 to ∞, we get
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.,0,0
zSzDdxxvzxD
(42)
Multiplying equation (33) by x and integrating over 0 to ∞, we get
.,0,0
zGzRdxxzxR
(43)
Multiplying equation (34) by xv1 and integrating over 0 to ∞, we get
.,0, 110
11 zVzQdxxvzxQ
Multiplying equation (35) by xv 2 and integrating over 0 to ∞, we get
.,0, 220
22 zVzQdxxvzxQ
(44)
Integrating equation (27) between 0 and ∞, we get
.,,3,2,1,
1,0 kj
zBzPzP
j
jj
(45)
Integrating equation (9.30) between 0 and ∞, we get
.
1,0
z
zSzDzD (46)
Integrating equation (33) between 0 and ∞, we get
.
1,0
z
zGzRzR (47)
Integrating equation (34) between 0 and ∞, we get
.
1,0
111
z
zVzQzQ (48)
Integrating equation (35) between 0 and ∞, we get
.
1,0
2111
z
zVzVzQzQ (49)
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Using equations (41) to (44) in equation (38), we get
k
m
m
k
m
jmmjj zGzSzPzpzBzPpzzP
11
.,0,0
.11 0,112 QzVzVp j (50)
Put 1j and 1k in equation (50), we get
zD
QzVzVzpzP
1
0,1121
1
11,0
(51)
where
zzBpzzzD 111
.zGzzSzGzzS
Putting 1j and 2k in equation (50), we get
0,13122111 ,0,0 QzzApzAzPpzAzP (52)
where
zzBpzzzA 111
zGzzSzGzzS
zGzzSzzBzA
22
zGzzS
.11 123
zVzVzA
Putting 2j and 2k in equation (50), we get
0,13221212 ,0,0 QzBpzBzPpzBzP (53)
where
zzBpzzzB 221
zGzzSzGzzS
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988
zGzzSzzBzB
12
zGzzS
.11 123
zVzVzB
Using equation (53) in (52), we get
zBzAppzBzA
zBzAzBzApQzpzP
222111
133220,11
1 ,0
(54)
where
zzzBzAzBzAp 13322
11 12
zVzV (55)
zzzzzBAppzBzA 222111
zzBp 11
zGzSzzGzzS
zzBp 22
.zGzSzzGzzS (56)
Using equations (55) and (56) in equation (54), we get
zD
QxVzVzpzP
2
0,1121
1
11,0
(57)
where
zzBpzBpzzzD 22112
.zGzSzzGzSz
Using equation (52) in (53), we get
zBzAppzBxA
zAzBzAzBpQzpzP
222111
133210,12
2 ,0
(58)
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989
.11 1213321
zVzVzzzAzBzAzBp (59)
Using equations (56) and (59) in equation (58), we get
.
11,0
2
0,1122
2zD
QxVzVzpzP
(60)
From equations (51), (54) and (60), we get
,
11,0
2
0,112
zD
QxVzVzpzP
j
j
kj ,,2,1 (61)
where
zGzzSzzzzD
3
k
m
mm zGzazSzBp
1
zDz
QzB
zVzVzp
zPj
j
j3
0,1
12
1
11
,
111
3
0,112
zD
QzBzVzVp jj
.,,2,1 kj (62)
Using equations (39) and (62) in equation (46), we get
k
m
mz
zSzpz
z
zSzDzD
1
11,0
zDz
zVzVzSz
3
12 111
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990
k
m
mm QzBp
1
0,1 .1 (63)
Using equations (39), (40), (42) and (62) in equation (47), we get
zDz
zVzVzSzGzzR
3
12 111
k
m
mm QzBp
1
0,1 .1 (64)
Using equation (36) in (48), we get
.
1
1
0,11
1 Qz
zVzQ
(65)
Using equation (36) in (49), we get
.
1
1
0,121
2 Qz
zVzVzQ
(66)
Adding equations (62) to (64), we get
k
m
m QzD
zNzRzRzDzP
1
0,14
1 (67)
where
zuzuzuzN 3211
11 121
zVzVzu
zSzGzzzu
12
k
m
mm zBpzu
1
3 1
.34 zDzzD
At zuz 1,1 becomes
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991
010011 121
VVu
000112
SGzu
k
m
mm
k
m
mm BpBpu
11
3 111
.01111 3211 uuuN (68)
Differentiating zN 1 with respect to z, we get 11 N and 1"1N as
01111,11111 321323211 uuuuuuuuN
111121111 "32323"21"1 uuuuuuuN
111111112 32"132321 uuuuuuuu
.1112 321 uuu (69)
Differentiating zu1 with respect to z, we get 11 u as
000011 12121
VVVVu
.21 VEVE (70)
Differentiating zu 2 with respect to z, we get 12 u as
000000112
SGSGSGu
.1 SEGE (71)
Using equations (68), (70) and (71) in equation (69), we get
.1121
1
212
"1
k
m
mm BpSEGEVEVEN
At zDz 3,1 and zD 4 becomes
k
m
mm GSBpGSD
1
3 000001
.0011 34 DD
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Differentiating zD 4 with respect to z, we get 14 D as
001011 334 DDD (72)
Differentiating zD with respect to z, we get 1"D as
.1212011 33"3"4 DDDD (73)
Differentiating zD 3 with respect to z, we get 13 D as
.111
11
3
k
m
mm
k
m
mm BpBpGESED (74)
Using equation (74) in (73), we get
.1121
11
"4
k
m
mm
k
m
mm BpBpGESED
Putting 1z in equation (67), we get
k
m
m QD
NRDP
1
0,14
1
0
0
1
1111 for m.
Using L’Hospital’s rule, we get
k
m
m QD
NRDP
1
0,14
1
0
0
1
1111 for m.
Again using L’Hospital’s rule, we get
k
m
m QD
NRDP
1
0,1"4
"1
1
1111
.
11
11
0,1
11
1
21
Q
BpBpSEGE
BpSEGEVEVE
k
m
mm
k
m
mm
k
m
mm
Adding equations (65) and (66), we get
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0,1
1221
1
11Q
z
zVzVzQzQ
0
0121 QQ for m.
So after using L’Hospital’s rule, we get
0,1
121221
1
1Q
zVzVzVzVzQzQ
.11 0,12121 QVEVEQQ (75)
Using the fact that
k
m m QQQRDP1 211 ,1211111 we get
210,1
VEVE
zXQ
(76)
where,
1zX (77)
where
.
11
1
1
k
m
mm
k
m
mm
Bp
BpGESE
(78)
Using equation (77) in (76), we get
210,1
1
VEVEQ
(79)
and SES
0 is the mean of delay time, SEG
0 is the mean of
repair time, 2211 0,0 VEVVEV are the mean of vacation times
of FRV and SOV respectively. 0,1Q is the steady state probability that the
system is idle due to server’s vacation. Also we have ,1 which is the
stability condition under which steady state solution exists.
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994
0,1
1
21 QzD
zNzQzQzRzDzPzP
k
m
m
zGzzzVzVzN
111 12
zzzBpzS
k
m
jj
1
1
zGzzSz
k
m
mm zGzzSzBp
1
zSzzGzzzVzV
11 12
k
m
mm zSzzGzzzBp
1
k
m
mm zGzzSzzBpzzz
1
zGzzS
zzzVzV
1111 12
k
m
mm zzzBp
1
1
,21 zuzu
where
11 121
zVzVzu
k
m
mm zBpzzu
1
2 1
zDzD 3
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zP is the probability generating function for the number of customers in
the queue.
4. Performance Measures
Let qL and L denote the steady state average queue size and system size
respectively.
Then
1
0,1
1
zz
q QzD
zN
dx
dzP
dx
dL
010011 121
VVu
k
m
mm
k
m
mm BpBpu
11
2 111 (80)
0111 21 uuN
k
m
mm GSBpGSD
1
3 000001
011 3 DD
0,1Q
zD
zN
dz
dL q
at 1z
0,12
Q
zD
zDzNzNzD at 1z
0,12
1
1111Q
D
DNND
0,12
1
1"11"1Q
D
DNND
.
12
1"11"10,12
Q
D
DNND
(81)
Differentiating zN with respect to z, we get 1N as
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996
11111 2121 uuuuN
.11 21 uu (82)
Differentiating zN with respect to z, we get 1"N as
11112111" "21212" uuuuuuN
.11211 212"1 uuuu (83)
Differentiating zu1 with respect to z, we get 11 u as
0000011 122111
VVVVVu
.21 VEVE (84)
Differentiating zu1 with respect to z, we get 1"1u as
.21 212
22
12
"1 VEVEVEVEu (85)
Differentiating zu 2 with respect to z, we get 12 u as
k
m
mm Bpu
1
2 11 (86)
Using equations (80) and (84) in equation (83), we get
.1
1
21
k
m
mm BpVEVEN
Using equations (80), (84), (85) and (86) in equation (83), we get
k
m
mm BpVEVEVEVEN
1
212
22
12
21"
.12 21
1
VEVEBp
k
m
mm
Differentiating zD with respect to z, we get zD as
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997
zDzD 3
11 3 DD (87)
zDzD "3"
11" "3DD (88)
Using equation (74) in (87), we get
.111
11
2
k
m
jj
k
m
mm BpBpGESED
Differentiating zD 3 with respect to z, we get 1"3D as
k
m
mm GESEBpD
1
2"3 1221
22
1 1
2212 GEGESESEBpBp
k
m
k
m
mmmm
.12
1
GESEBp
k
m
mm
(89)
Using equation (89) in (88), we get
k
m
mm
k
m
mm BpGESEBpD
11
221221"
22
1
2 GEGESESEBp
k
m
mm
GESEBp
k
m
mm
1
12
where 2
22
122
,,, VEVEGESE are the second moments of delay, repair,
FRV and SOV time respectively. L can be obtained using the relation
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998
. qLL Using Little’s formula, we obtain ,qW the average waiting time
in the queue and W, the average waiting time in the system, as
q
q
LW and
LW respectively.
5. Particular case
If there is only one service 1k and no breakdown 0 then, we
get
211
12 111
VEVEzBz
zVzVzP
which coincides with the probability generating function obtained by Gautam
Choudhury irrespective of the notations used.
6. Numerical illustration
To illustrate the effect of some parameters on the system performance
measures, we present a numerical example by considering service times and
vacation times as exponentially distributed as follows.
Assuming the values 12,9,3.0,5.0,2.0,3 21321 pppk
1,5,55.0,5.1 21 vvv for the parameters, subject to the stability
condition and varying the values of from 0.43 to 0.53 in steps of 0.01 and
from 1 to 2 in steps of 0.1 we have calculated the values of ,qL the
corresponding graph is drawn in figure 1. From figure-1, one can notice that
the surface displays an upward trend as expected for increasing value of the
arrival rate and SOV probability against the average queue size .qL
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999
Figure-1.
We consider service times and vacation times are distributed as Erlang-2
distribution. Assuming the values ,7,3.0,5.0,2.0,3 1321 pppk
1,5,51,1,5,8 2132 vvv for the parameters, subject to the
stability condition and varying the values of from 0.21 to 0.28 in steps of
0.01 and from 1 to 1.14 in steps of 0.02 we have calculated the values of ,qL
the corresponding graph is drawn in Figure-2. From Figure-2, we see that the
surface displays an upward trend as expected for increasing values of the
arrival rate and SOV probability against the average queue size .qL
Figure-2.
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References
[1] Y. Baba, On the 1GMX queue with vacation time, Operation Research Letters 5
(1986), 93-98.
[2] A. Badamchi Zadeh and G. H. Shankar, A two phase queue system with Bernoulli
feedback and Bernoulli schedule server vacation, Information and Management Sciences
19 (2008), 329-338.
[3] A. Borthakur and G. Chaudhury, On a batch arrival Possion queue with generalised
vacation, Sankhya Ser.B 59 (1997), 369-383.
[4] G. Chaudhury, An 1GMX queueing system with a set up period and a vaction period,
Questa 36 (2000), 23-28.
[5] B. D. Choi and K. K. Park, The 1GM retrial queue with Bernoulli schedule, Queueing
Systems 7 (1990), 219-228.
[6] M. Cramer, Stationary distributions in a queueing system with vaction times and limited
service, Queueing Systems 4 (1989), 57-68.
[7] B. T. Doshi, Queueing systems with vacations-a survey, Queueing Systems 1 (1986), 29-
66.
[8] B. T. Doshi, Conditional and unconditional distributions for 1GM type queues with
server vacation, Questa 7 (1990), 229-252.
[9] S. Fuhrmann, A note on the 1GM queue with server vactions, Questa 31 (1981), 13-
68.
[10] B. R. K. Kashyap and M. L. Chaudhry, An introduction to Queueing theory, Kingston,
Ontario, 1988.
[11] J. Keilson and L. D. Servi, Oscillating random walk models for 1GG vacation systems
with Bernoulli schedules, Journal of Applied Probability 23 (1986), 790-802.
[12] Y. Levi and U. Yechilai, An sMM queue with server vactions, Infor. 14 (1976), 153-
163.
[13] K. C. Madan, On a 1bMMX queueing system with general vacation times,
International Journal of Information and Management Sciences 2 (1991), 51-61.
[14] K. C. Madan, An 1GM queue with optional deterministic server vacations, Metron, 7
(1999), 83-95. 1GM Feedback queue with two types of service 33
[15] K. C. Madan and R. F. Anabosi, A single server queue with two types of service,
Bernoulli schedule server vacations and a single vacation policy, Pakistan Journal of
Statistics 19 (2003), 331-342.
[16] K. C. Madan and G. Choudhury, A two stage arrival queueing system with a modified
Bernoulli schedule vacation under N-policy, Mathematical and Computer Modelling 42
(2005), 71-85.
Page 27
ANALYSIS OF A MULTI TYPE SERVICE OF
Advances and Applications in Mathematical Sciences, Volume 20, Issue 6, April 2021
1001
[17] E. Rosenberg and U. Yechiali, The 1GMX queue with single and multiple vacations
under LIFO service regime, Operation Research Letters 14 (1993), 171-179.
[18] H. Takagi, Queueing Analysis: A foundation of performance evalution, Vol 1, North
Holland, Amsterdam, 1993.
[19] H. Takagi, Time dependent process of 1GM vacation models with exhaustive service,
Journal of Applied Probability 29 (1992), 418-429.