published online june - july 2016 in ijeast ( ...tesma108,ijeast.pdf · serial and non serial...

International Journal of Engineering Applied Sciences and Technology, 2016 Vol. 1, Issue 8, ISSN No. 2455-2143, Pages 344-355 Published Online June - July 2016 in IJEAST (http://www.ijeast.com)

344

ANALYSIS OF GENERAL QUEUING

MODEL OF MIXED QUEUES WITH

BALKING AS WELL AS RENEGING

DUE TO LONG QUEUE AND SOME

URGENT MESSAGE WITH FINITE

WAITING SPACE

Dr Meenu Gupta,

Assistant Professor Dept of Mathematics,

Dr. B. R. A. Govt. College, Kaithal, 136027 (Haryana) India

Dr. Man Singh,

Ex Prof., CCS, Haryana Agricultural University, Hisar, India

Dr. Deepak Gupta,

Prof. & Head, Dept. of Mathematics,

M. M. Engg. College, Mullana (Ambala) – India

Abstract: This paper considers the most appropriate

& more general queuing model in respect of

customers which are allowed to leave the system at

any stage with or without getting service. This

queuing model for infinite waiting space has been

derived earlier and with reference to that queuing

model for finite waiting space has been derived in this paper. The paper considers the steady-state

behavior of the queuing processes when M service

channels in series, are linked with N non-serial

channels having balking & reneging phenomenon,

wherein:

Each of M service channels has identical

multiple parallel channels.

Poisson arrivals & exponential service times are

followed.

The service discipline follows SIRO rule (service

in random order) instead of FIFO rule (first in first out).

The customer becomes impatient in the queue

after sometime and may leave the system without

getting service and can renege due to urgent

message.

Waiting space is finite.

Keywords: Poisson stream, Reneging, Balking,

Traffic intensity, Steady-state, Parallel channels,

Urgent message.

I. INTRODUCTION

Singh and Ahuja (1996) worked on serial and non

serial queuing processes and obtained the numerical

solutions for mean queue length. The solutions of

serial and non serial queuing processes with reneging

and balking phenomenon have been studied by

Vikram and Singh (1998). The steady state solution of serial and non serial queuing processes with

reneging and balking due to long queue and some

urgent message and feedback phenomenon is

obtained by Singh, Punam and Ashok (2008). In our

present society, the impatient customers generate the

most appropriate and modern models in the theory of

queues. Incorporating this concept, we study the

steady state analysis of general queuing system in

this chapter in the sense that:

M service channels in series are linked with

N non-serial channels having reneging and

balking phenomenon where each of M


345

service channels has identical multiple

parallel channels.

The input process is Poisson and the service

time distribution is exponential.

The service discipline follows SIRO-rule

(Service in random order) instead of FIFO-

rule (first in first out).

The customer becomes impatient in queue

after sometime and may leave the system without getting service and can renege due

to some urgent message also.

The input process depends upon the queue

size in non-serial channels.

Waiting space is finite.

II. FORMULATION OF MODEL

The system consists of Q i ( i =1,2,….M) service

phases where each service phase Qi has ci ( i=1,2,….M) identical parallel service facilities and

Q1j channels (j=1,2,….N) with respective servers Si

(i=1,2,……M) and S1j (j=1,2,……N).Customers

demanding different types of service arrive from

outside the system in Poisson distribution with

parameters λ i( i =1,2,….M) at Qi service phase and

λ1j(j=1,2,…..N) at Q1j service phase respectively .

But the sight of long queue at Q1j , may discourage

the fresh customers from joining it and may decide

not to enter the service channel Q1j (j=1,2,….N) then

the Poisson input rate λ 1j would be 1

1

j

jm

where mj

is the queue size of Q1j . Further, the impatient

customers joining any service channel may leave the

queue without getting service after a wait of certain

time. The service time distribution for the server Si

(i=1,2……,M) and S1j (j=1,2…..,N) are mutually independent negative exponential distributions with

service rates ( 1,2,........., )i i M and

1 ( 1,2,...., )j j N respectively. After the

completion of service at Qi (i=1,2,……..M), the

customer either leaves the system with probability pi

or joins the next phase with probability qi such that pi

+qi =1 (i= 1,2…..M-1). After completion of service at

QM , the customer either leaves the system with

probability Mp or joins any of the Q1j

(j=1,2,…….,N) with probability 1j

Mj

m

q

(j=1,2,….,N) such that

1 1

NMj

M

j j

qp

m

=1.

If the customers are more than ci in the Qi service

phase ,all the ci servers will remain busy and each is

putting out the service at mean rate i and thus the

mean service rate at Qi is ci i ,on the other hand if

the number of customers is less than ci in the Qi service phase ,only ni out of the ci servers will be

busy and thus the mean service rate at Qi is ni i

(i=1,2,…….M).It is assumed that the service commences instantaneously when the customer

arrives at an empty service channel.

Here we assume that if at any instant, there are K

customers in the system

1 1

M N

i j

i j

n m K

,

then the customers arriving at that instant will not be

allowed to join the system and it is considered as a

forced balking and lost for the system.

The customers may leave any service channel

without getting service if they receive urgent call

while waiting.


346

III. FORMULATION OF EQUATIONS

Define P(n1 ,n2 , ……….nM ; m1 ,m2,m3, …………mN ;t) as

the probability that at time ‘t’ , there are n i customers ( which may renege or after being serviced by the Qi

phase either leave the system or join the next service

phase ) waiting in the Qi service phase

(i=1,2,……..,M) , mj customers ( which may balk or

renege or after being serviced leave the system )

waiting before the servers S 1j (j=1,2,…….N) .

We define the operators T i , T i and T i , i+1

to act upon the vector 1 2( , ,............., )Mn n n n

and T j and T . j and T j , j+1 to act upon the

vector 1 2( , ,............., )Nm m m m as follows:

T i ( n ) = (n1,n2,………ni-1…nM)

T i ( n ) = (n1,n2,………ni+1…nM)

T i , i+1 ( n ) = (n1, n2 ,….,ni +1, ni+1 -1,….,nM )

T j ( m ) = (m1,m2,………mj-1…mN)

T . j ( m ) = (m1,m2,………mj+1…mN)

T . j , j+1 ( m ) = (m1 ,m2 ,……,mj +1,mj+1. -1….mN

)

Following the procedure given by Kelly (1979), we

write difference differential equations as:

1 1 1 1

( , , ) ( ) ( , ; )1 i i i i j

M N M Nij

i i in n c i in j i j j jm

i j i jj

dP n m t n b r m R P n m t

dt m

1

1 1

( , ; ) ( , ; )M N

j

i i j

i j j

P T n m t P n T m tm

1

1 1 , 1

1 1

( ) ( , ; ) ( , ; )i i i i

M M

n c i in i i in i i

i i

b r P T n m t q P T n m t


347

1 1 1 2

1 1

( , ; ) ( , , , 1, ; )i M

M NMj

i in i Mn M j

i j j

qp P T n m t P n n n T m t

m

1 1

1

( , ; )j

N

j j jm j

j

R P n T m t

-------------------------------------------------------------- (1)

for 0in , 0jm and 1 1

M N

i j

i j

n m

< K; (i = 1,2,3,…,M; j=1,2,3,…,N)

and

1 1

( , ; ) ( ) ( , ; )i i i i j

M N

i in n c i in j ij j jm

i j

dP n m t n b r m R P n m t

dt

1

1 1

( , ; ) ( , ; )M N

j

i i j

i j j

P T n m t P n T m tm

1

1 , 1

1

( , ; )i

M

i in i i

i

q P T n m t

1 1 2

1

( , , , 1, ; )M

NMj

Mn M j

j j

qP n n n T m t

m

----------------- (2)

for 0in ; 0jm and 1 1

M N

i j

i j

n m K

; (i= 1, 2, 3, …, M; j=1, 2, 3, …, N)

Where 1 0

( )0 0

when xx

when x

( )

0

1i i

i i

n c

i i

when n c

when n c

0

0

01

01

1

1

; 1, 2,......

(1 )

; 1, 2,......

(1 )

i

i i

i

Ti iini

T jj

j

T jj j

m j

i i i i

in

i i i i

T

n

i

in

m

j

jm

n when n c

c when n c

er i M

e

eR j N

e


348

Where i jin jmr and R are the average rates at which

the customers renege after a wait of certain time

0 0i jT and T whenever there are ni and mj customers

in the Qi and Q1j service phases respectively. ib and

j are the mean reneging rates at serial and non-

serial service phases due to some urgent message and

( , ; ) 0P m n t if any of the arguments is negative.

IV. STEADY-STATE EQUATIONS

We write the following Steady-State equations of the

queuing model by equating the time derivatives to

zero in the equations (1) and (2)

1

1

1 1 1 1

( ) ( , )1 i i i i j

M N M Nj

i i in n c i in j j j jm

i j i jj

n b r m R P n mm

1

1 1

( , ) ( , )M N

j

i i j

i j j

P T n m P n T mm

1

1 1 , 1

1 1

( ) ( , ) ( , )i i i i

M M

n c i in i i in i i

i i

b r P T n m q P T n m

1

1

( , )i

M

i in i

i

p P T n m

1 1 2

1

( , , , 1, )M

N

Mn M j

j

P n n n T m

1 1

1

( , )j

N

j j jm j

j

R P n T m

-------------------------------------------------------------------------------- (3)


;M N

i j

i j

n m K

and

1

1 1

( ) ( , )i i i i j

M N

i in n c i in j j j jm

i j

n b r m R P n m

1

1 1

( , ) ( , )M N

j

i i j

i j j

P T n m P n T mm

1

1 , 1

1

( , )i

M

i in i i

i

q P T n m

1 1 2

1

( , , , 1, )M

NMj

Mn M j

j j

qP n n n T m

m

----------------------- (4)


349


;M N

i j

i j

n m K

Two cases arise depending upon the number of customers in and number of channels ic at iQ phase

(i= 1, 2, 3, …, M)

V. CASE(1)

When the number of customers ni before iQ phase is

less than the number of identical service channels ic

(i.e. in < ic ;i = 1, 2, 3,…, M) , then there is no

reneging in iQ and the service is immediately

available to the customers on arrival. Then under

such situation i in c

= 0 and iin i in

Steady State Equations:

The equations (3) and (4) reduce to

1

1

1 1 1 1

( , )1 j

M N M Nj

i i i j j j jm

i j i jj

n m R P n mm

1

1 1

( , ) ( , )M N

j

i i j

i j j

P T n m P n T mm

1

, 1

1

( 1) ( , )M

i i i i i

i

q n P T n m

1

( 1) ( , )M

i i i i

i

p n P T n m

1 2

1

( 1) ( , , , 1, )N

M M M j

j

n P n n n T m

1 1

1

( , )j

N

j j jm j

j

R P n T m

--------------------------------------------------------------------------------------- (5)


;M N

i j

i j

n m K

and

1

1 1

( , )j

M N

i i j j j jm

i j

n m R P n m

1

1 1

( , ) ( , )M N

j

i i j

i j j

P T n m P n T mm


350

1

, 1

1

( 1) ( , )M

i i i i i

i

q n P T n m

1 2

1

( 1) ( , , , 1, )N

M M M j

j

n P n n n T m

-----------(6)


;M N

i j

i j

n m K

Steady State Solutions:

The Steady State solutions of the above equations (5) and (6) can be verified to be

1 2'

1 2 1 1

1 1 2 2

1 1, 0,0

n n

qP n m P

n n

3' '

3 2 2 1 1

3 3

1 1Mn n

M M M

M M

q q

n n

1 2

1 2

11 1 12 2

1 2

11 1 1 12 2 2

1 1

1 1m m

M M M M M M

m m

j j

j j

q q

m mR R

1

1

1

1N

N

m

N M MN M

m

N

N N Nj

j

q

mR

----------------------------------------------- (7)

Where ' ' ' '

1 2 3 1, , , M and M are as mentioned below:

'

1 1

'

1 1

' '

1 1 ; 2,3,........, 1

M M MM

M

k k k k

q

q k M

VI. CASE (II)

When the number of customers before iQ phase is

more than or equal to the number of identical service

channels ic (i.e. i in c ), then there is reneging in

iQ (i=1,2,…,M). So under such situation 1i in c

and iin i ic

and ( ) 1in .


351

Steady State Equations:

The equations (3) and (4) reduce to

1

1

1 1 1 1

( , )1 i j

M N M Nj

i i i i in j j j jm

i j i jj

c b r m R P n mm

1

1 1

( , ) ( , )M N

j

i i j

i j j

P T n m P n T mm

1

1 , 1

1 1

( ) ( , ) ( , )i

M M

i in i i i i i i

i i

b r P T n m q c P T n m

1

( , )M

i i i i

i

p c P T n m

1 2

1

( , , , 1, )N

M M M j

j

c P n n n T m

1 1

1

( , )j

N

j j jm j

j

R P n T m

-------------------------------------------------------------------------------------- (8)

for 0jm and 1 1

;M N

i j

i j

n m K

and

1 1

( , )i j

M N

i i i in j ij j jm

i j

c b r m R P n m

1

1 1

( , ) ( , )M N

j

i i j

i j j

P T n m P n T mm

1

1 , 1

1 1

( , ) ( , )i

M M

in i i i i i i

i i

r P T n m q c P T n m

1 2

1

( , , , 1, )N

M M M j

j

c P n n n T m

---------------------------------------------------------------- (9)

for 0jm and

1 1

;M N

i j

i j

n m K


352

Steady State Solution:

The Steady state solution of the above equations can be verified to be:

2

1

1

1 22

1

2 1 1 1 1 1 1 1 1 11

1 1 1 1 2 2 2 2 1 1 1 1

1 1

, 0,0

nn

n

n nn

i i i n

i i

c b r c qP n m P

c b r c b r c b r

3

33

2

3 1 2 2 2 2

1

2

3 3 3 3 1

1 1

i

i

n

i i i in

i

n n

i i i i in

i i

c b r c q

c b r c b r

1

1 1 1 1 1

1

1

1

1 1

M

i

MM

i

nM

M i i i in M M M M

i

n M n

M M M Mi i i i in

i i

c b r c q

c b r c b r

1

1

'

11 1

1 11 1 1

1

m

M M M M

m

j

j

c q

m R

2

2

'

12 2

2 12 2 2

1

m

M M M M

m

j

j

c q

m R

'

1

1

1

...............................................

N

N

m

M M MN M M

m

N N N Nj

j

c q

m R

------------- - (10)

Where 1 2 3 1, , , M and '

M are as mentioned below:

1

1 1 1 1 1

' 1

1

1 1

1

( )

( ) ( )

i

M i

M

M i i i in M M M M

i

M M

M M M Mn i i i in

i

c b r c q

c b r c b r


353

1 1

1

1 1 1 1 1

1

( ) ; 2,3,..........., 1.i

k

k k i i i in k k k k

i

c u b r q u c k M

Here, it is mentioned that the customers leave the

system at constant rate as long as there is a line

provided that the customers are served in the order in

which they arrive. Putting jjm jR R (j=1,2,3,…,N)

in equations (5), (6),(8) and (9) and iin ir r in

equations (8) and (9), the steady-state solutions (7)

and (10) reduce to

1 2'

1 2 1 1

1 1 2 2

1 1, 0,0

n n

qP n m P

n n

3' '

3 2 2 1 1

3 3

1 1Mn n

M M M

M M

q q

n n

1 2

11 1 12 2

1 11 1 1 2 12 2 2

1 1...............

m m

M M M M M Mq q

m R m R

1

1

1............................... (11)

Nm

M M MN M

N N N N

q

m R

for i in c , m j ≥ 0 ; (i=1,2,3,……,M) ; j=1,2,3,……..N).

Where ' ' ' '

1 2 3 1, , , M and M are the same as mentioned before.

1 2

2 1 1 1 1 1 1 1 11

1 1 1 1 1 1 1 1 2 2 2 2

, 0,0

n n

c b r c qP n m P

c b r c b r c b r

32

3 2 2 2 2

1

3

1

n

i i i i

i

i i i i

i

c b r c q

c b r

1

1 1 1 1

1

1

MnM

M i i i i M M M M

i

M

i i i i

i

c b r c q

c b r


354

1'

11 1

1 11 1 1

1m

M M M Mc q

m R

2'

12 2

2 12 2 2

1m

M M M Mc q

m R

'

1

1

1..................

Nm

N M MN M M

N N N N

c q

m R

--------------------------------------------------------------

(12)

for i in c , m j ≥ 0 ; (i=1,2,3,……,M ; j=1,2,3,……..N) .

Where 1 2 3 1, , , M and '

M are the same as mentioned before.

We obtain 0,0P from the normalizing condition

0 0

, 1K K

n m

P n m

and 1 1

M N

i j

i l

n m K

and

with the restrictions that the traffic intensity of each

service channel of the system is less than unity.

It has been verified that the results obtained in case

of finite waiting space are confirmative with the

results of infinite waiting space.

It shows that our results obtained for finite space in

equation no. 12 are in confirmation with our

corresponding results already published in 2011 for

infinite space, inspite of change in different

differential equations.

VII. REFERENCES

1. O’Brien, G.C. (1954): The solutions of some queuing problems.

J.Soc. Ind. Appl. Math. Vol. 2, 132-

142.

2. Jackson, R.R.P. (1954): Queuing

systems with phase type service.

Operations Research Quarterly. Vol.

5, No.2, 109-120.

3. Barrer, D.Y. (1955): A Waiting

line problem characterized by

impatient customers and indifferent

clerk. Journal of Operations Research.

Society of America. Vol.3, 360-367

4. Hunt, G.C. (1956): Sequential

arrays of waiting lines. Opps. Res. Vol.

4, 674-683.

5. Finch, P.D.(1959): Cyclic Queues

with Feedback. J. Roy Statist. Soc. B-

21, 153-157.

6. Maggu, P.L. (1970): On certain

types of queues in series. Statistica

Neerlandica. Vol. 24, 89-97.

7. Singh, Man.(1984): Steady-state

behavior of serial queuing processes

with impatient customers. Math.

Operations forsch .U. Statist. Ser,

statist. Vol. 15, No.2. 289-298.

8. Gupta, S.M.(1994):

Interrelationship between queuing

models with Balking and Reneging

and Machine repair problem with

warm spares. Microelectron, Reliab.,

Vol. 34, No. 2, pp. 201-209.

9. Singh. Man and Singh, Umed

(1994): Network of Serial and non-

serial queuing processes with impatient

customers. JISSOR Vol. 15, pp. 1-4.

10. Singh Man and Ahuja Asha (1996): The Steady State Solutions of Multiple Parallel Channels in Series

With Impatient Customers. Intnl. J.

Mgmt. Sys. Vol. 12, No. 1 .67-76.

11. Singh Man , Punam and Ashok

Kumar (2008): Steady-state solutions

of serial and non-serial queuing

processes with reneging and balking

due to long queue and some urgent

message and feedback . International

Journal of Essential Science Vol. 2,

No.2. 12-17.

12. Punam; Singh, Man and Ashok


355

Kumar (2011): The Steady –State

solution of serial queuing processes

with reneging due to long queue

and urgent message. Applied Science

Periodical Vol.XII, No. 1.

13. Singh, Satyabir and Singh, Man

(2012): The Steady-state Solution of

Serial channels with Feedback and

Balking connected with Non-Serial

Queuing Processes with Reneging

Balking. Global Journal of Theoretical

and Applied Mathematical Sciences.

Vol. 2, No. 1, pp 1-11.

14. Gupta, Meenu and Singh, Man

and Gupta, Deepak (2012): The

steady-state solutions of multiple

parallel channels in series and Non-serial multiple parallel channels both

with balking &reneging. International

journal of computer application,

Issue 2, Vol. 2, pp 119-126.

published online june - july 2016 in ijeast ( ...tesma108,ijeast.pdf · serial and non serial...

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