STEADY STATE SOLUTIONS FOR A FINITE CAPACITY QUEUEING
SYSTEM WITH DETERMINISTIC SERVICE TIMES
Hambeer Singh
Department of Mathematics, R.I.E.T., Faridabad, Haryana
Abstract
In the paper an attempt has been made to compute some characteristics of M/D/1 queue with finite capacity. A closed form formulae for the distribution of the number of customers in the system has been derived. An explicit solution for mean queue length and average waiting time has also been provided, Embedded Markov Chain have been used for computing in this model.
Keywords: Embedded Markov Chain. Generating function, Coefficient of Z-transform, Steady stat solutions, Mean queue length, average waiting time.
1. Introduction:
In the present scenario, a number of servicing systems are providing services with constant servicing time. In the present paper, a model has been studied whose arrival pattern is in Poisson fashion and service facility is deterministic with finite capacity. M/D/1 queue is by far the simplest and the most general model. The model has a variety of applications not only in the telecommunication area, but also in the area of operation research, computer science and many other engineering areas. The model has been studied for a long time and is solved from a computational point of view.
Artalejo & Corral (1997) analyzed steady state solution of a single-server queue with linear repeated requests. Ahahiru sule Alfa and Gordan J. Fitzpatrick (1999) considered a Geo/D/1 queue operating under a hybrid FIFO/LIFO discipline and obtain the waiting time distribution. Bruneel & Kim (1993) gave a discrete time models for communication systems including ATM. Erlang (1909) studied a queueing system with Poisson arrival and deterministic i.e. constant, service time. This model is appropriate for continuous deterministic service time queueing systems, input can be seen as completely random or as a super position of the large number of processes. This follows from the fact that in situations with many sources each having a small generation rate.
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Franx, G.J. (2001) gave a simple solution for the M/D/1 waiting time distribution”. Gravey,Louvion et al. (1990) studied on the Geo/D/1 and Geo/D/1/N. Kumar, R and Sharma, S.K. (2012) studied M/M/1/N queueing system with relation of reneged customers. Such type of models was also studied by Roberts, Mocci and Virtamo (1996), Vicari and Trangia (1996), Brun and Garcia (2000) gave an analytical solution of finite capacity M/D/1 queues. Woodward (1994) studied communication and computer networks, modeling with discrete time queues.
2. Notations & Terminology:
: Arrival rate
T : Service time of each customer is the same and constant
: Utilization factor = T
Pj(N) : Probability of j-customer
Aj,l : Expected amounts of time that I-customer is present during a service time that is started with j-customers in the system.
XN(t) : Number of customers in the system at time t
tn : The date of the nth customer departure
qj (tn) : Probability those j-customers are left behind by the nth departure
l : probability of l arrivals during a customer service
A(Z) : Z-transform of the sequence
TN : Average system time
WN : Mean waiting time
3. The Model
In this section a model has been studied which has finite capacity with deterministic service time. In the queueing system there are N-1 places in the waiting room. The service mechanism is FCFS i.e. “First Come First Served” discipline. Some customers, who upon arrival see a full system are rejected, that is called lost customer.
Let Aj,I the expected amount of time that I customers are present during a service time that is started with j-customers in the system. A scheme for the time average probabilities
P-j(N) is known for the model M/G/1/N, we get
l j l j j ll N P N A p N A
P
1
, ,
1
0( ) ( )
)
( 1 l N ... (1 )
Since arrivals are Poisson distributed with rate and service time is deterministic i.e. constant T, we have
e
l
l l
! ... (2 )
Also probability transition matrix of the Imbedded Markov Chain, and the stationary distribution
0 0 3 -N 0 2 -N 0 2 -N 0 -1 .... ... ... ... 0 0 0 . . . . . . . . . . . . . . . -1 ... ... ... ... 0 -1 ... ... ... ... -1 ... ... ... ... l 3 N 1 0 l 2 N 2 1 0 1 2 N 2 1 0... (3 )
Q = [q0, q1 ...qN - 1], is a ei g en vector of t he m atrix, so
Q = Q ... (4)
2 0 2 1 0 1 2
2 3 0 2 1 1 2 0 2 1 2 0 1 1 0 1 0 1 0 0 0 .... ... .. ... ... ... ... ... .. ... ... ... ... ... .. ... ... ... ... ... N N N
N q q q q
q q q q q q q q q q q q
... (5 )
This is a li near s yst em of N -1 equations involving the N -unknowns q0, q1 ,
-- -- -- -- -- -- -- -- -- -- -- -- qN - 1 , qn=anq0, then
a0=1, a1=e-1 and that a2 ………..aN - 1
obe y t he following recursion :
1 01
1
1 a a
a e a n n i i n i n
n
... (6 )
Let A (z) be t he z - trans form
0 ) ( l l lz a zA ... (7 )
4. Generating Function A(Z)
Multipl y b y Zn and s umming over equati on (6), we get
Let A(Z) be the Z -t rans form
1 1 0 0 0 0
n i i i n n n n n n nnz a a z a a z
a ... (8 )
Using equati on (2) we ge t
) (z
A = n
n n n i i i n z z a z n z e
a
0 0 1 0 1 ! ) ( =
0 0 0 0 ! ) ( 1 n n i i i z e n z a z a z e ea
) (z
A = ( ( ) 0)
) 1 ( ) 1 (
0 A z a
z e e
a
z
z
[ 0 ( ) 0]
) 1 ( a z A za z e z
(1 )
1 )
(z ze z
A = 1z
Where, B(z) =
(1 )
1 1 z ze 5. The Coefficients of Z-Transforms (bn)
We have n
n n n n n n n
nz b z z b z
a
0 0 0 =
1 1 1 0 n n n n nnz b z
b b
0 0 10 ( )
n n
n n n n
nz b b b z
a ... (10)
We can writ e
0 1 0 n n n n n
nz a a z
a ... (11)
B y equation (10) and (11), we get
1 1 10 1 ( )
n n
n n n n
nz b b z
a
a ... (12)
Which proves that a0 = 1, an = bn – bn - 1
B y the z -transform F(z) as follows
n l n n l l n l l z e l n l z
F
0 0 ) ( ) ( ! ) 1 ( )
( ... (13)
In equation (13), nl m, we have
m l l l m m l l z z e m l z
F
0 0 !
) 1 ( )
( ... (14)
Since l l mz
m l l e z m l
0 ! ) 1 (Adjusting thi s val ue in equation (14)
0 ) 1 ( ) ( m m z m z e zF ... (15)
If e(1z)z 1,then the s eries F(z) converges .
If z < 1, we have the following expression for the z -t rans form F (z)
) ( 1
1 )
( (1 ) B z
ze z
F z
Hence we get ( ) , 1 !
) 1 (
0
)
(
n e
l n l b
n
l
l l n l l
n
... (16)
B y the P robabilit y norm aliz ation
, 1
1
0 1
N
l
q Thus
1 1
0 0
1 1
N Nl l
b a
q ... (17)
and qn = anq0
,
1 1
N n n n
b b b
q n=1, 2, ... N -1 ... (18)
6. Steady State Solution
The steady state probability distribution Q of the number of customers left behind by a departure differs from the queue length distribution.
P = [P0(N), ... . PN (N)], i n M/ D/1
queue wi th fini te capacit y.
We know that t he probabilit y of j -cust om er in cas e of M/G/1/ N queues
, ) ( 1
) (
N P
N P q
N j
j ... (19)
j = 0, 1, 2 ... N -1
A sim pl e conservati on l aw hol ds t hat
1 ( )
1 ) ( 1
( P0 N
T N
PN
... (20)
using the res ult = T
(1PN(N)
1
1P0(N)
(1PN(N)
=
1
1 ( )
0
1
q N PN
With the hel p of equation (17), we have,
) (
1PN N =
1
1 ) ( 1 1 1
N
b N P
Arrangi ng above equation, we get
1 1
1 ) ( 1
N N N
b b N
P
... (21)
B y equation (19), ot her probabiliti es are gi ven b y
1 0
1 1 )
(
N
b N
P
... (22)
1 1
1 ) (
N j j j
b b b N P
... (23)
Using equati on (21) we have t o cal culat e mean number of cust omers.
7. Mean Queue Length and Average Waiting Time
XN be the m ean num ber of cust om ers of the M/D/ 1/N queue. Since, we know that
N
l l N lP N
X
0
)
( ... (24)
We have
1 1
0
1 1
1 ) (
N N
l
N l
l N
b Nb P
P l N X
... (25)
Solving above equati on, we get mean num bers of custom er s
1 1
0
1
N N
l l N
b b N
X
... (26)
Let TN be t he average s ys t em ti me, b y Li t tle’s l aw
XN = (1 -PN(N)) TN ...(27)
B y equation (27)
1 P (N)
X TN N
N ... (28)
1 1
0 1
1 1
1 1
1
N N
l l N
N N N
b
b b
N N
b b T
1 1
0
N N
l l N
b N b N
T T
... (29)
We know that,
Waiting time = Average system time – Service time
i.e. WN = TN – T = T
b N b N
T
N N
l l
1 1
0
) (
Average waiting time
T b
N b N
W
N N
l l N
1 1
0
1
... (30)
8. Conclusion:
In the preceding sections, the expressions for mean queue length and average waiting time have been obtained. A closed form formula for the distribution of the number of customers in the M/D/1 queue capacity has also been provided.
9. References:
[1] Ahahiru, S.A. and Fitzatrick, G.J. (1999): “Waiting time distribution of a Fifo/ Lifo M/D/1 queue”, INFOR, 37, 149-159.
[2] Artalejo,J.R. and Gomez-corral,A (1997): “Steady state solution of a single-server queue with linear repeated requests”, J.Appl. Probab. 34,223-233.
[3] Brun, O. Garcia J.M. (2000): “Analytical solution of finite capacity M/D/1 queues”, Journal of Applied Probability 37 (4).
[5] Erlang, A.K. (1909): “Probability and telephone calls”, Nyt. Tidsskr Mat. Ser.B., 20, 33-39.
[6] Franx, G.J. (2001): “A simple solution for the M/D/1 waiting time distribution” Operation Research, 29, 221-229.
[7] Gravey,A.,Louvion, J.R.,and Boyer,P (1990): “On the Geo/D/1 and Geo/D/1/N queues”, performance Evaluation 11(2).
[8] Kumar, R and Sharma,S.K. (2012): “ M/M/1/N queueing system with relation of reneged customers.” Pakistan ournal of Statistics and Operation Research Vol.8, no.4,pp 859-866.
[9] Vicari,N.and tran-Gia, P.(1996): “A numerical analysis of the Geo/D/N queueing system”, Research report No.151,Institute of Computer Science ,University of Wurzburg.
[10] Woodward (1994), M.E.:“Communication and computer networks:Modeling with discrete-time queues”, CA:IEEE computer society Press, Los Alamitos.
