Transient analysis of an infinite server queue with catastrophes and server failures Dr.S.Sophia Murali T.S. SSN College of Engineering 1
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Apr 13, 2017
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Transient analysis of an infinite server queue with catastrophes and
server failures
Dr.S.Sophia Murali T.S.
SSN College of Engineering1
A Transient analysis
Practically usable solution
Catastrophe, server
failures
Analytically tractable
model
State dependent
queues
LAYOUT OF PRESENTATION
MODEL DESCRIPTION
SOLUTIONMETHODSADOPTED
ANALYSISOF DIFFERENT CASES
DEVELOPMENT OF MODEL
First In-First Out (FIFO)Infinite waiting room - Arrival rate with n customers in the
queue - Service rate with n customers in the
queue - Catastrophe rate (Poisson’s distribution) - Failure probability at time t - State dependent probability with n
customers at t - Repair rate (Exponential distribution)Let be the number of customers at
time t Without loss Of generality
nn
)(tQ
)(tPn
1)0(0)0(
0
PQ
RttX ,
Chapman Kolmogorov Difference differential Equations
)()()()()(1100
0 tQtPtPdttdP
)()()()()(1111 tPtPtP
dttdP
nnnnnnnn
)](1[)()( tQtQdttdQ
,....3,2,1n
1......
3......
2......
Infinite server Queue rates
,....3,2,1,,....2,1,0,
nnn
n
n
SOLUTION METHOD ADOPTED
Taking Laplace Transform of We get from 1
From 2
From 3
3,2,1
)()(*
sssQ
)()(
10
)(*0
*0
*1)(
1)(
sPsP
ss
ssP
)()(
1
1*1
*
*
*1)()(
)(
sPsP
nnn
n
n
n
n
nssPsP
4......
6......
5......
From repeated iteration of using 6 and incorporating infinite server queues
Continued fraction approachRepresenting as Kummer function and using the
identity
(also known as Hypergeometric function)
)(*0 sP
...)(1
)()2(
2)(
)(*0
ss
ss
ssP
....)2()2(
)1()1()(
);1;1();;(
11
11
zc
zazczazc
zcaFzcaFc
We get
Now, by iterating 6 as continued fraction and representing as Kummer function,
And using the same identity…
sF
sPs
ss
);1;1()1()( 11)(
*0
);;()1();1;1(
)()(
1
11*1
*
nnFnsnnF
sPsP
s
s
n
n
Obtaining a Generalized expression forwe get…
Converting the Kummer function to its definition using the Pochhammer symbol defined by
And observing that
)(* sPn
,....3,2,1,0,))...(2)()((
);1;1()1()( 11)(*
n
nssssnnF
sPsn
ssn
0
11 !)()();;(
k k
kk
kczazcaF
k.
1);;0(11 zcF
1),1)...(1(
0,1
kkaaaa
ka
k
k
It becomes
On simplifying, it yields
By partial fraction expansion,We have
))...(2)()((!)1(
)()1()1(
)( 0)(
*
nsssskn
n
sP k ks
kkn
ss
n
0
0
)(* ,....2,1,0,
)(!!
)!()1()1()(k
kn
l
knk
ssn nlskn
knsP
On inverting, we get
Inverting we get,
0 0
)(*
)()!(!)1(
!!)!()1()1()(
k
kn
i
i
kn
knk
ssn isikniknknsP
)1(
!))1((
)(te
ntt
n enee
tP
dueenee ut
te
nuuu
)1(!
))1((
0
)1(
]1[)( tetQ
)(* sQ
IT ALSO VERIFIES THAT SUM OF PROBABILITIES IS 1
0
1)()(n
n tQtP
STEADY STATE PROBABILITY DISTRIBUTIONThe steady state failure distribution is obtained
as
The steady state probability
In particular
Q
ssQs
)(lim *
0
n
n
nsP
nnnF
ssP
))...(2)(();1;1(
lim 11*
0
);1;1(110
sFP );1;1(1 110
FQP
Also,
On expanding and simplifying and puttingThe above equation reduces to
This is the well known result from the Gross and Harris. Absence of catastrophes – remains the same for both and queues.
0 0 )()!(!)1(
!!)!()1(
k
kn
i
i
kn
knk
n iikniknknP
0
,....3,2,1,0,
!
nne
Pn
n
,....3,2,1,0,!
nne
Pn
n
nP1//MM //MM
Steady state probability generating function
Substituting for and , we get
Using the identity,
00 n
n
n
nn QzzPz
0
11
))...(2)(();1;1(
n
n
nnnF
z
yxcaFxncnaFncya
n n
nn
;;;;!)(
)(1111
0
nP Q
Solving using the definition of the Pochhammer symbol
And rewriting
Substituting in
On differentiating and taking limit, we get mean
which is the average number of customers in system at steady state
1...321 nn
n
nn 1))...(2)((
z
0
11
1);1;1(
n nn
n nnFzz
z
XE
zz
z 1lim
Using
We get the second moment as
P[server is busy]= =
P[server is idle or under repair]= =
XEzzXE
z
2
2
1
2 lim
)2)((
2 22XE
1nnP
1
11
))...(2)(();1;1(
1n
n
nnnF
Q
QP 0 );1;1(1 11
FQQ
P[server is busy/server is up]= =
P[server is idle/server is up]= =
Q
Pn
n
11
1
11
))...(2)(();1;1(
n
n
nnnF
QP10 );1;1(11
F
ReferencesM. Abramowitz and I. Stegun, Handbook of
Mathematical functions, Dover, New York(1965).B. Krishna Kumar, A. VijayaKumar, and S.Sophia,
Transient analysis of a Markovian queue with chain sequence rates and total catastrophes, Int. J. Oper. Res.,5, No. 4(2009),375-391.
L. Lorentzen and H. Waadeland, Continued fractions with applications, North-Holland, Amsterdam(1992).
D. Towsley and S.K. Tripathy, A single server priority queue with server failures and queue flushing, Oper. Res. Lett., 10, No.6 (1991), 353-362.
Thank you
inedussnmechmurali ...@15054SSNCEMurali ..ST
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