Analysis of M/M/c/N Queuing System With Balking, Reneging and Synchronous

Vacations

Dequan YueDepartment of Statistics, College of Sciences

Yanshan University, China

Wuyi YueDepartment of Information Science and Systems Enginee

ring Konan University, Japan

Outline

Introduction System Model Analysis

1. Steady-state equations

2. Block matrix solution method

3. Steady-state probabilities

4. Special cases Conditional Distribution Conclusions

Introduction

Practical background Literature review Purpose of our research

Practical Background

Balking and reneging - Customers who on arrival may not join the queue when

there are many customers waiting ahead of them

- Customers who join the queue may leave the queue without getting serviced

- A common phenomenon in many practical queuing situations

- Examples in communication systems , production and inventory system, air defence system and etc., see, Ancker and Gafarian (1962), Shawky(2000) and Ke (2007)

Practical Background (cont.)

Server vacations -In many real world queuing system, server may become

unavailable for a random period of time -Server vacations can represent the time when the server is

performing some secondary task -Single server queuing models with vacations have been

studied by many researchers and applied in many fields such as communication systems, production and inventory system, computer networks and etc, see, Doshi(1986,1990), Takagi(1991) and Tian and Zhang(2006)

-There are a few studies on multi-server vacation queuing system in the vacation model literature

Literature Review

Three types of the processes 1. Birth and death (BD) process

2. Quasi birth and death (QBD) process

3. Level-dependent quasi birth and death (LDQBD) process

Literature Review (cont.)

BD process model

1. M/M/c queue with balking and reneging (Montazer-Haghighi et.al. (1986))

-infinite buffer and a constant probability of balking,

2. M/M/c/N queue with balking and reneging (Abou-El-Ata and Hariri (1992))

-finite buffer and a state-dependent probability of balking ,

In these models, the state processes are classical BD processes. The steady-state probability have been obtained by solving the difference equations

Literature Review (cont.)

QBD process model

1. M/M/c queue with synchronous vacations (Tian et. al. (1999))

2. M/M/c queue with synchronous vacations of partial server ( Zhang and Tian (2003))

In these models, the state processes are infinite QBD proces

ses. The steady-state probability have been obtained by using matrix geometric solution method (Neuts (1981))

Literature Review (cont.)

LDQBD process model

-M/M/c /N queue with balking, reneging and server breakdowns (Wang and Chang (2002))

- M/M/c /N queue with balking, reneging and synchronous vacation of partial servers (Yue et al. (2006))

In these models, the state process are finite LDQBD processes. The closed form expressions for steady-state probabilities have not been obtained

Purpose of Our Research

To study an M/M/c/N queue with balking, reneging and synchronous vacation

To present a closed form expression for computing the steady-state probability of the model

To present a block matrix method for solving a special queuing model with LDQBD process

To show that some existing models in the literature are special cases of our model

System Model Customers arrive according to Poisson process with rate Service time is assumed to be exponentially distributed with rate A customer who on arrival finds customers in the system, either

decides to enter the queue with probability or balks with probability

After joining the queue, a customer reneges if its waiting exceeds a certain time which is assumed to be exponentially distributed with rate

All the servers take synchronous vacation when the system is completely empty. The vacation time is assumed to be exponentially distributed with rate

nb1 nb

n

rT

c

Procedure of Analysis

Step 1. Develop steady-state equations

Step 2. Partition infinitesimal generator and steady-state

probability vector

Step 3. Rewrite steady-state equations in block matrix

form

Step 4. Find solution in block matrix form

Step 5. Compute inversions of some block matrices

Step 6. Find the closed form expressions of steady-state

probabilities

Analysis Notation the number of customers in system at time servers are taking on vacation at time servers are not taking on vacations at time is a Markov process with state space:

Note: If we label the states in lexicographic order:

then the Markov process is a level-dependent QBD Process

t( )L t

0,( )

1,J t

t

t{ ( ), ( ); 0}L t J t t

{( ,0) : 0,1,..., } {( ,1) : 1,2,..., }i i N i i N

),0,0(

)1,(),0,(...,),11(),0,1(),0,0( NN

Analysis (cont.)

0 0

1 1 1

2 2 2

1 1 1N N N

N N

B C

A B C

A B CQ

A B C

A B

Steady-State Equations

Notation Steady-state probabilities:

Steady-state equations

where is steady-state probability vector, is a generator, and is a column vector of order

0 ( ) lim ( ( ) , ( ) 0}t

P n P L t n J t

1( ) lim ( ( ) , ( ) 1}t

P n P L t n J t

0 1 0 1 1 1( (0), (1),..., ( ), (1), (2),..., ( ))P P P P N P P P N

0

1

PQ

Pe

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

(2 1)N

Block Matrix Solution Method Partitioned block structure

where

11 12 13

22 23

31 33

0 1

0

1

Q Q Q N

Q Q Q

Q Q N

N N

0 0 1( , ( ), )P P P N P

0 0 0 0( (0), (1),..., ( 1))P P P P N

1 1 1 1( (1), (2),..., ( ))P P P P N

Block Matrix Solution Method (cont.)

The steady-state equations in block matrix form

where is a column vector of order with all its

components equal to one

0 11 1 31

0 12 0 22

0 13 0 23 1 33

0 0 1

0,

( ) 0,

( ) 0,

( ) 1N N

PQ PQ

PQ P N Q

PQ P N Q PQ

Pe P N Pe

Ne N

Solution in Block Matrix Form

Theorem 1.

The segments of the steady-state probability vectors are given by

where

10 0 22 12

1 11 0 23 22 12 13 33

( ) ,

( )( )

P P N Q Q

P P N Q Q Q Q Q

1 1 1 10 22 12 23 22 12 13 33( ) {1 ( ) }NP N Q Q e Q Q Q Q Q

Computing Inverse Matrix

Notations,1,...,2,1,0, Nibu ii

,,...,2,1, Niivi

,,

1,...,2,1,

Niv

Niuvw

N

iii

,,...,2,1,)(

,...,2,1,

Nccicic

ciisi

1,...,1,,

1,...,2,1,

Nccib

cit

ii

Computing Inverse Matrix Lemma 1-Inverse of matrix For the elements of the matrix is given by

where are given by the following recursive relations

where and

1,2,..., ,j N 112Q

0,ijc

1

1,ij

j

cu

1

1,ij ij

j

c ku

1 11 2

1 1

,i iij i j i j

i i

w vk k k

u u

1,2,..., 1,i j

,i j

1, 2,...,i j j N

1, 2,...,i j j N

ijk

1jjk 1 0j jk

12Q

Computing Inverse Matrix (cont.)

Explicit expression of

where is a row vector of order 2 and

ijk

1 1... ,Tij j j ik A A A 1, 2,..., ,i j j N 1,2,...,j N

1

,

0

k

kk

k

k

w

uA

v

u

(1,0)

1,2,..., 1k N

Example 1

12

0.9167

1.8333 0.8333

1.0000 2.2500 0.7500

1.5000 2.667 0.6667

2.0000 3.0833 0.5833

2.5000 3.5000 0.5000

Q

112

1.0909

2.400 1.2000

5.7455 3.6000 1.3333

17.5818 11.7000 5.3333 1.5000

73.2338 49.5000 23.6190 7.9386 1.7143

424.7273 288.0000 138.6667 748.000 12.0000 2.0000

Q

Computing Inverse Matrix (cont.)

Lemma 2- Inverse of matrix For the elements of matrix is given by

the empty summation is defined to be zero.

1,2,..., ,j N 133Q

1 1

1 1

11 1

1 1

...,

...

... 1,

...

i k k j

k k k j

ijj

k k j

k k k j j

t t t

s s sd

t t t

s s s s

1,2,..., 1,i j

, 1,...,i j j N

0

1k

33Q

Example 2

33

3.000 2.000

2.000 4.000 2.000

3.000 4.5000 1.5000

4.5000 5.8333 1.3333

6.0000 7.1667 1.1667

7.5000 7.5000

Q

133

1.0000 1.0000 0.6667 0.2222 0.0494 0.0077

1.0000 1.5000 1.0000 0.3333 0.0741 0.0115

1.0000 1.5000 1.3333 0.4444 0.0988 0.0154

1.0000 1.5000 1.3333 0.6667 0.1481 0.0230

1.0000 1.5000 1.3333 0.66

Q

67 0.3148 0.0490

1.0000 1.5000 1.3333 0.6667 0.3148 0.1823

Steady-state Probabilities

Theorem 2 The steady-state probabilities are given by

where

10 ( ) ,jP j

0,1,..., 1,j N

0

1( ) ,P N

1

1 11

( ) ( ),N

Nj ij ii

P j d d

1,2,...,j N

1

11 1 1

1 ( ),N N N

j Nj ij ij j i

d d

1 ,

,

N N j N N j

jN N N

v c w c

w c

1,2,..., 1,j N

j N

Special Cases

M/M/c/N queue with balking and reneging Let and

then our model becomes the model studied by Abou-EI-

Ataharir (1992)

1,

(1 ( 1) ),

( 2)n

m

b n c N

n c

0,1,..., ,n c

, 1,...,n c c N

Special Cases (cont.)

M/M/c queue with balking and reneging Let and

then our model becomes the model studied by Haghighi et al. (1986)

M/M/c queue with synchronous vacation Let and then our model

becomes the model studied by Tian et al. (1999)

1,

,nb

0,1,..., ,n c, 1,..., ,n c c N

,N 0 1,ib 0,1,...,i

Conditional Distribution

Let represent the conditional queue length given that

all servers are busy, then we have

Theorem 3

The conditional stationary distribution of the queue length is given by

where and are given in Lemma 2 and Theorem 2

cQ

1

( ) ( ) 11

1

11

( ) ,( )

N

N j c i j c ii

c N N

Nj ij ij c i

d dP Q j

d d

0,1,...,j N c

ijd j

Conditional Distributions (cont.)

Remark 2. Based on Theorem 2, we can obtain some other performances such as the expected number of customers in the system and in the queue, etc. However, these performance have very complex expressions.

Conclusions Studied an M/M/c/N queue with balking , reneging and

synchronous vacations Presented a block matrix method for solving the steady-

state equations Presented a closed form expressions for steady-state

probabilities Obtained some existing models in the literature as special

cases of our model Derived the conditional distributions for the queue length

Thank you very much!