reservicing failed customers under - · pdf filehas the laplace transform (feller 1966), the...

International Journal of Statistics and Systems

ISSN 0973-2675 Volume 12, Number 3 (2017), pp. 593-606

© Research India Publications

http://www.ripublication.com

Reservicing Failed Customers under three Disciplines

with - Poisson Arrivals

Sheeja S. S.1 and Dr. V. R. Saji Kumar2

1Research Scholar, Dept. of Statistics, Manonmaniam Sundaranar University, Abhishekapatti, Tirunelveli-12, India.

2Associate Professor, Dept. of Statistics, Christian College, Kattakada, Thiruvananthapuram, Kerala, India.

Abstract

A single server non-Markovian queuing system with – Poisson arrivals and

general service time distribution is discussed, in which a customer may fail with

probability p (0< p< 1). The failed customers are reserviced under the following

three disciplines (i) reservices a failed item immediately after completion of the

service of the customer in the main queue (MQ) if the item served has failed (ii)

failed items are stockpiled in a failed queue (FQ) and reservices only after all

customers in MQ are served and (iii) the server turned to FQ when the stockpile

reaches a threshold (N). The probability generating function (PGF) of the size of

the system at the moment of departure of each customer in the main queue and the

mean busy period are obtained.

Key words: - Poisson Arrivals, Half Cauchy Distribution, Mittag – Leffler

Function.

1. INTRODUCTION

As a waiting time distribution, the Mittag-Leffler probability law introduced by Pillai

(1990) has considerable applications in Physics (Weissman et al. 1989, Weron et al.

1996), economics (Manardi et al. 2000, Sabatelli et al. 2002), insurance mathematics

(Kozubowski 1999), queuing theory, renewal theory, computer networks and

replacement of electronic components (Pillai 1988, 1993).

594 Sheeja S.S. and Dr. V.R. Saji Kumar

In the above cited situations some items may fail and require reservice, especially in

the replacement of electronic components. Pillai (1988, 1993), Pillai and Sabu

George (1984) and Anil (2001) have shown that there are many situations especially

in reliability problems and replacement of electronic components require thick tailed

distributions to model and the Mittag-Leffler distribution provide alternative models

in these situations where exponential distribution won’t work. Thus we have the

following.

When the inter-arrival time distribution in an evolutionary process is Mittag-Leffler

with Laplace transform

s ( 0,10,0 s ), the corresponding

arrival process {N(t), t 0} is - Poisson and has the probability density function

(Anil 2001, Saji Kumar 2003)

))(()( ntNPtPn

= )1)((

)()1(

)(

0

nkt

nnk nk

k

k

, n = 0, 1, 2, ..... (1.1)

when =1, (1-1) is the probability density function of the usual Poisson distribution

with parameter t

For h > 0

)()()1()()()(

1 hotPntPnth

tPhtPnn

nn

where )(ho are terms containing

h and higher powers of h

hence )()()1()( 1 tnPtPntPtnnn

(1.2)

when ,1 (1.2) is same as XVII, (2.7) in Feller (1968)

In the analysis to follow we denote the Laplace transform of a function by the same

function with an asterisk and argument S, the PGF with a caret and argument u ,

unless otherwise specified

REMARK 1.1

A random variable X is said to have an - Poisson distribution with parameter if

and only if it has the probability density function

,)1)((

)1()()(

0

xkxxk

xPxk

k

k

,.......2,1,0x

Reservicing Failed Customers under three Disciplines with α-Poisson Arrivals 595

)(xP has the PGF, the Mittag-Leffler function

10)),1(()(ˆ uuEuP (1.3)

where

0

)1()(k

k kuuE

)(xP is the discrete version of the - inverted stable distribution, the first passage

time distribution of the -stable Levy motion (Saji Kumar 2002, 2003), the

corresponding density function is

0),sin(!

)1()1(

1)( 1

1

1

ykyk

kyg kk

k

k

)(yg has the Laplace transform (Feller 1966), the Mittag-Leffler function

)()(* sEsg

In sequel we frequently use the half Cauchy distribution with a special

reparametrization, which is the ratio of two one sided identically distributed positive

stable random variables having the Laplace transform )exp( s . The respective

probability density function and distribution function (Saji Kumar, 2002).

0,0;cos2

sin)(

22

1

yyy

yyh

and

cos

sintan

1)( 1

y

yH (1.4)

)(yh has the Laplace transform, the Mittag-Leffler function (Saji Kumar 2002)

)1(

)()1()(

0

*

kssH

k

k

k

= ))(( sE (1.5)

We consider a single server Non-Markovian queuing system in which the customers

arrive according to an -Poisson law with mean )1(

, where some customers

must be reserviced with probability p . These models arise naturally in a production


line, in sending message through the network, replacement of electronic components,

communication problems and other fields (Pillai 1988, 1993 and references there in)

and thus -Poisson distribution have a wide range of applications. When messages

are sent through a network, some may be returned and must be sent again. In a

telephone centre, some calls may not be completed due to signal failure and must be

reestablished. In an assembled unit of electronic components, some items must be

replaced again. In such situations as an inter-arrival time distribution, the Mittag-

Leffler probability law better fits the data than the exponential distribution (Pillai

1988, 1993).

Optimal operating policies for M/G/1 system in which the server may be turned on

and off have been studied over a period of more than three decades and an active area

of research. Earlier authors include Yadin et al. (1967), Sobel (1969), Bell (1971),

Yechiali (2004), Salehi-Rad et al. (2004). Following the reservicing facility discussed

in Salehi-Rad et al. (2004), we extend the analysis and gets new results for the

queuing system with -Poisson arrivals and Mittag-Leffler inter-arrival time

distribution.

The purpose of this paper is to establish the importance of the Non-Markovian arrival

process, that is, the -Poisson arrivals and the Mittag-Laffler inter-arrival time

distribution in the realm of queuing theory. The model is a generalization of the

M/G/1 queuing model, in the sesce that when =1, it is the usual classical M/G/1

model with reservicing facility discussed in Salehi-Rad et al. (2004).

The paper is organized as follows. In section 2 we describe the mathematical model

of the system, in section 3, 4 and 5, we describe the three reservicing discipline I, II

and II respectively.

2. MATHEMATICAL DESCRIPTION

As mentioned earlier we consider a single server non-Markovian queuing system with

-Poisson arrivals and general service time in the study state in which some

customers are failed with probability p and require reservice. We consider three

alternative procedures for reservicing the failed customers.

(i) the server reservices a failed item immediately after completion of the

service of the customers in the main queue, if the item served has failed

(ii) failed items are stockpiled in a failed queue and reserviced only after all

customers in MQ are served. After completion of reservice of all items in

FQ, the server returns to MQ if there are customers waiting, otherwise the

system is idle.

(iii) it is same as (ii) except that the server starts reservice if there are N failed

items in FQ (threshold N). Again all items in FQ are reserviced before

returning to MQ


Let t1, t2,....., be the service completion epoch of customers in MQ. The service

time(s) and reservice time ( s ) are assumed to be independent and have the general

distribution B1 and B2 respectively. 1/b1 and 1/b2 denote the respective mean of B1

and B2. )(sAn and )(sAn denote the number of arrivals in MQ during servicing in MQ

and reservicing in FQ at the moment of n th departure in MQ.

Let Xn denotes the number of customers remaining in MQ at the service completion of

the n th customer.

};0{1)()()1( 111 nnnnn XIsAsAXX (2.1)

In discipline II and III ),( nn YX denote the number of customers remaining in MQ

and FQ at the service completion of the n th customer. For discipline II

,,1)()()1(),( 1}0{}0{

1

1111

nXnX

Y

iinnnnn UIYIsAsAXYX

nn

n

(2.2)

and for discipline III

1

1

)(11 1332,)()()1(),( nCn

Y

iCUCCinnn UIYIIsAsAXYX

n

(2.3)

where (X-1) is max {X-1, 0}, 11 nU , if the departure has failed and is zero

otherwise,

}0,0{1 NYXC nn , },0{2 NYXC nn ,

}0,0{3 NYXC nn and }0,0{4 nn YXC

2.1 Queuing Discipline I

In this section we discuss discipline I and obtain the PGF and the mean busy period.

The following lemmas are required for finding the PGF

LEMMA 3.1

If X is a non-negative integer-valued random variable with PGF )(uP , then

)]0()1()(ˆ[1

)( )1( XPuuP

uuE X

For proof see Salehi-Rad et al. (2002)


LEMMA 3.2

If )(sA and )(sA are the number of arrivals during service and reservice times in MQ,

then their PGF’s are

))1(()(1*

11 uMuQ ,

))1(()(1*

22 uMuQ ,

Where, *

1M and *

2M are the Laplace-Stieltje’s transform of the product M1=B1H and

M2=B2H respectively. H is given in (1.3)

PROOF

)()1)((

)()1()( 1

)(

0 0 0

1 tdBnk

tn

nkuuQ

nk

k n

nk

= )()1(

)()1( 1

0 0

tdBn

tkn

nu

n

k kn

knk

= )()1()1(

)()1( 1

00 0

tdBukn

nt k

n

k

kn

n

n

= )()1()1(

)()1( 1

0 0

tdBun

t nn

n

n

= )())1(( 1

1

0

* tdButH

Where H* is defined in (1.5)

By Parseval relation (Feller 1966, pp 437)

))1(()(1*

11 uMuQ , where

*

1M is the LST of the product M1 = B1H

Similarly

))1(()(1*

22 uMuQ , where

*

2M is the LST of the product M2 = B2H

When =1, Q1(u) and Q2(u) are identical with Q1(u) and Q2(u) in Salehi-Rad et al.

(2004)


LEMMA 3.3

By Lemma 3.2,

)()0)((( *

2 MsAPE

LEMMA 3.4

For -Poisson arrivals the LST of the busy period satisfies the functional equation

)())(1()()(0

tdBestEs st

PROOF

Customers arrive at a server according to the -Poisson law with parameter . The

successive service times are assumed to be independent random variables with

common distribution B. On the arrival of a customer at epoch o , the server is free

and his service time commences immediately, the customers arriving during his

service time join a queue and the service time continue without interruption as long as

a queue exists. The busy period is the time interval from zero to the first epoch when

the server becomes free. Its duration is a random variable, its distribution function

and Laplace transform are respectively denoted by B and .

Let the first customer depart at epoch t and N be the number of customers joined the

queue during his service time. N is an -Poisson variable with parameter . The

total service time required by the N customers

SN = X1+X2+......................+ XN,

Where each Xj have the Laplace transform . The total duration of the service times

has the same distribution as the busy period NSt

By Feller (1966, pp 448-449) and using (1.3)

0 0

)()1(

))(1()()1()(

n

stn

nn tdBen

sts

=

0

)())(1()(( tdBestE st (3.1)

which completes the proof.

When, =1, (3.1) is same as (4.1) in Feller (1966, pp 448-449)


THEOREM 3.1

The PGF of nX in the steady state is

01

*1

*

1

*

2

1*

1

)1()1(()1(

))(1(1)1(()1()( P

uupcuMP

MpuMuuP

(3.2)

Where c* is the LST of the convolution of the distribution function M1 and M2

respectively and P0 is the probability that the MQ is empty and

))(1(1

1*

2

0

MpP

(3.3)

Where )()1(

)()1(

21

BEpBE

(3.4)

PROOF

Taking the PGF of (2.1) and using lemmas 3.1, 3.2 and 3.3, leads to (3.2)

When, 1 , (3.2) and (3.3) is same (3.2) and (3.3) in Salehi-Rad et al. (2004)

REMARK 3.1

The service time of an individual customer is the sum of the service time in MQ and

p times the service time, if it required in FQ. Thus the mean service time

21

111bpbb

Thus we have a queuing system with -Poisson arrivals and general service time

distribution B with mean service time b1 . The mean busy period is obtained by

taking the derivative of (3.1) with respect to s .

E (busy period) = )()1(

)1(1

BEb

,

1

11

b

Where )1(

)(

BE


2.2 Queuing Discipline II

As mentioned in the introduction here we have a single server, general service time

queuing system with -Poisson arrivals, in which the failed customers in FQ are

reserviced when all customers in MQ (MQ is empty) are served and again switches to

MQ if there are customers in MQ. Xn denotes the number of customers in MQ and Yn

denotes the number of failed customers in FQ at epochs }{ nt . To find the joint PGF

P(u,v) we require the following lemma.

LEMMA 4.1

The PGF of the number of served customers (departures) in a busy period (here in

MQ)

PROOF

)(1()( *

1 uuMu (4.1)

By Feller (1966, pp 449-450) and using (1.3)

)()(1)1(

)()1()( 1

0 0

tdBuk

tuu k

k

kk

)())(1()( 1

0

tdButEu

1*

1 ))(1( uuM

When 1 , (4.1) is same as Feller [1966, XIV, (10.3)]. See also Feller (1966 pp

449-450), Takacs(1955) and Saaty(1961).

THEOREM 4.1

The joint PGF of (Xn, Yn) in the steady state is

01

*

1

**1

*

1

)1()1(

),,()1(),,,()()1()1(

),( PuuMpvp

poGupuGvRuMpvpvuP

(4.2)

Where,

0,)0()( jvXjYPvR jnn


))1((1(),,,(1

*

2

* upMppuG (4.3)

1*

1 )(1)( uuMu

0P is the probability that MQ is empty

)0(0 nXPP

= ),,()1(

)1(*

12

2

1

poGp

(4.4)

)()1(

11

BE

and )(

)1(22

BE

PROOF

Taking the joint PGF of (2.2) we get (4.2) using L’Hospital’s rule and the fact that Lt

P(u,1)=1 as u 1, we may get 0P .

Note that the number of failed items in FQ is distributed as Binomial with parameters

p and K . K is the number of served customers and has the PGF )(u . By solving

(4.3) and using the PGF of the binomial distribution we can find R(v), the PGF of nY ,

given nX =0 in terms of )(u . Then mean busy period

E(Busy period) = E(Busy period in MQ) + PE (Busy period in FQ)

= 1211 1

11

1

11

bp

b

= )1(21

21

bbbpb

2.3 Queuing Discipline III

Here the server turned to FQ if the store is full (threshold N) or if the MQ is empty.

After completion of service in FQ the server returns to MQ if the MQ is non-empty.

The following remark establishes the probability that the FQ reaches the threshold N.

REMARK 5.1

When the store reaches its maximum (threshold N) the server turned to FQ. At this

time the number of departure from MQ is a random variable D, distributed as negative

binomial


,.......1,,)1(1

1)(

NNdpp

Nd

dDP dN

We use the notation NP. to represent it

That is )()(. dDPNYPP nN

THEOREM 5.1

The joint PGF of (Xn,Yn) in the stead state is

uuMpvp

PuRuMv

PpGupuGvRuMpvp

vuPNN

NN

NN

1*

1

1*

2

.0

**

1*

1

)1()1(

.)()))1(((

,,0()1(),,,()(

)1()1(

),( (5.1)

Where

jN

jnn vXjYPvR )0()(

0

NYIUEuR nxX

N n

n }0{)(

0, iU i

Ni

)( NYiXP nnN

i (5.2)

0),,,()(

* 1

n

sA

N XuEpuGnY

ii

=

))1((

1*

2 uMR

)0())((

.1.

2

*

2

2

0

nn

N

XYEMRP

P

PROOF

Taking the joint PGF of (2.3) and using the relations between indicator function.

We get (5.1)

To find 0P , use L’Hospital’s rule and Lt P(u,1) = 1 as 1u


SPECIAL CASE

As N , 0))1((*

2 NN uMv and

),,,(),,,( ** puGpuGN resulting in discipline II

That is

uuMpvp

PpGupuGvRuMpvpvuP

1*

1

0

**1

*

1

)1()1(

,,0()1(),,,()()1()1(

),(

Here the total busy period can be split up into four sub-periods

(T1) When the service commence in MQ with one customer

(T2) When the server turned to FQ when the threshold is N

(T3) When the server turned to MQ from FQ after reservicing all customers in FQ,

if there are items in MQ. This time there are

N

iinn sAXV

1

)(

waiting customers in MQ, Xn are remaining customers when the threshold is N,

N

iisA

1

)( are new arrivals during reservicing in FQ

(T4) when the server again turns to FQ after servicing all customers in MQ, now FQ

contains }),min{,....2,1( *KNYY nn customers,

Where K* is the number of departures in MQ when the server starts with Vn

customers.

E(Busy period ) = 111

2

21

)1(1

)1( bp

NNbN

pbN

+ }),{(1 *

2

KNMinYEEb n

}),min{( *KNYE n = NP, if min(N,K*) = N

= pE(K*), otherwise

Where E(K*) = b1(E(T3) and ),()1(

11

BE

)(

)1(22

BE

Proof follows from Salehi-Rad et al. (2004).


3. CONCLUSIONS

the single server non-Markovian queuing system with

– Poisson arrivals and general service time distribution. The probability generating

function (PGF) of the size of the system at the moment of departure of each customer

in the main queue and the mean busy period are obtained.

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reservicing failed customers under - · pdf filehas the laplace transform (feller 1966), the...

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