analysis of m/m/c/n queuing system with balking, reneging and synchronous vacations dequan yue...
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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
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Outline
Introduction System Model Analysis
1. Steady-state equations
2. Block matrix solution method
3. Steady-state probabilities
4. Special cases Conditional Distribution Conclusions
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Introduction
Practical background Literature review Purpose of our research
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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)
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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
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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
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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
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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))
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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Thank you very much!