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