UDC 004.4
Analysis of Two-Heterogeneous Server Queueing
System
∗†
H. Okan Isguder
∗
∗
, U. U. Kocer
Department of Statistics,
Dokuz Eylul University,
Tinaztepe Yerleskesi, Izmir, 35390, Turkey
†
Department of Statistics,
Dokuz Eylul University,
Tinaztepe Yerleskesi, Izmir, 35390, Turkey
This study mainly concerned with the K -capacity queueing system
with recurrent input and two heterogeneous servers. Arriving customers choose
server from the empty servers with equal probability. At an arrival time the customer joins the queue if both servers are busy. In addition, an arrival leaves
without service when the system capacity is achieved. The dened system is
represented by semi-Markov process and embedded Markov chain is obtained.
Steady-state probabilities are found and loss probability is calculated by analyzing stream of overows.
Abstract.
embedded Markov chain, heterogeneous servers, nite capacity queue,
loss probability, stream of overows.
Keywords:
1.
Introduction
Heterogeneous server queueing models were rst studied by Gumbel
(1960). Lin and Elsayed (1978), Neuts and Takahaski (1980), Fakinos
(1980), Nawijn (1983) Pourbabai and Sonderman (1986), Alpaslan and
Shahbazov (1996) Isguder, Kocer and Celikoglu (2011) are some examples
of the literature on heterogeneous server queueing models. The analysis
of GI/M/n/n queueing loss system was presented by Isguder and Kocer
(2014). They analyzed the stream of overows and obtained LaplaceStieltjes transform of the distribution of stream of overows are presented.
In addition, they formulated the steady-state probabilities by using embedded Markov chain of the semi-Markov process. In this study, by using the
results given by Isguder and Kocer (2014), the loss probability is calculated
for the GI/M/2/K queueing system. This study is an extension of Isguder
and Kocer (2014) in that, it is assumed in this study there is a waiting
space and there are two servers. Model description and assumptions are
introduced in the following section. In section 3, the semi-Markov process
representing the system is constructed and the analysis of the model is presented. The steady-state probabilities and loss probability are obtained in
section 4. Finally discussion and conclusions are presented.
2.
Model description and assumptions
Let X(t) be the number of customers at time t and Xn = X(tn −0), n >
1. Xn is the number of customers being in the system at the time of the nth arrival. The semi-Markov process that represents this system is dened
as follows:
ξ(t) = Xn , tn 6 t < tn+1 , n > 1.
The kernel of the process {ξ(t), t ≥ 0} is
Qij (x) = P {(Xn+1 = j, tn+1 − tn < x)|Xn = i},
(1)
for all x > 0 and 0 ≤ i, j ≤ K . For each state i, j the kernel function given
by (1) can be written as follows:
Z
1 x
[(1 − e−µ1 t ) + (1 − e−µ2 t )]dF (t),
(2)
Q00 (x) =
2 0
Z
1 x −µ1 t
(e
+ e−µ2 t )dF (t),
(3)
Q01 (x) =
2 0
Z x
Q10 (x) =
(1 − e−µ1 t )(1 − e−µ2 t )dF (t),
(4)
0
Z
Q11 (x) =
x
[e−µ1 t (1 − e−µ2 t ) + e−µ2 t (1 − e−µ1 t )]dF (t),
(5)
0
Z
Q12 (x) =
x
e−(µ1 +µ2 )t dF (t),
(6)
0
for i + 1 ≥ j ≥ 2 and i ≥ 2
Z x
[(µ1 + µ2 )t]i+1−j −(µ1 +µ2 )t
e
dF (t),
Qij (x) =
(i + 1 − j)!
0
for i + 1 > 2 and j = 0
Z xZ
Qij (x) =
[
0
0
t
[(µ1 + µ2 )t]i+1−2 i−2 −(µ1 +µ2 )y
y e
(i − 2)!
(7)
(8)
(1 − e−µ1 (t−y) )(1 − e−µ2 (t−y) )dy]dF (t),
for i + 1 > 2 and j = 1
Z xZ t
[(µ1 + µ2 )t]i+1−2 i−2 −(µ1 +µ2 )y −µ1 (t−y)
[
Qij (x) =
y e
[e
(i − 2)!
0
0
(1 − e−µ2 (t−y) ) + e−µ1 (t−y) (1 − e−µ1 (t−y) )]dy]dF (t),
(9)
Qij (x) = 0, j ≥ i + 1,
(10)
Let qij (s) represent the Laplace-Stieltjes transform of Qij (x) such that:
Z ∝
qij (s) =
e−sx dQij (x), 0 ≤ i, j ≤ K(Re{s} ≥ 0).
0
Hence q(s) = [qij (s)]K
0 is obtained in matrix form as follows:


q00 (s)
q01 (s)
0
0
···
0


 q10 (s)

q11 (s)
q12 (s)
0
···
0


 q (s)

q21 (s)
q22 (s)
q23 (s)
···
0
 20

q(s) = 

.
.
.
.
.
.


..
..
..
..
..
..




qK−1,0 (s) qK−1,1 (s) qK−1,2 (s) qK−1,3 (s) · · · qK−1,K (s)
qK−1,0 (s) qK−1,1 (s) qK−1,2 (s) qK−1,3 (s) · · · qK−1,K (s)
(11)
From (11) q(s) is a lower Hessenberg matrix. Let the transition probabilities are dened as pij = P {Xn+1 } = j|Xn = i and P = [pij ]K
0 . Hence pij
for each i, j can be obtained from the equation pij = qij (0).
The entries of the matrix given by (11) can be obtained by using
Laplace-Stieltjes transforms of the functions (2), (3), (4), (5), (6), (7),
(8), (9), (10) for the GI/M/2/4 queueing system as follows:
1
[2f (s) − f (s + µ1 ) − f (s + µ2 )],
2
1
q01 (s) = [f (s + µ1 ) + f (s + µ2 )],
2
q10 (s) = f (s) − f (s + µ1 ) − f (s + µ2 ) + f (s + µ1 + µ2 ),
q00 (s) =
q11 (s) = f (s + µ1 ) + f (s + µ2 ) − 2f (s + µ1 + µ2 ),
q12 (s) = f (s + µ1 + µ2 ),
(µ1 + µ2 )2
(µ1 + µ2 )
]f (s + µ1 + µ2 ) −
f (s + µ1 )
µ1 µ2
µ2
(µ1 + µ2 )
−
f (s + µ2 ) + (µ1 + µ2 )f 2 (s + µ1 + µ2 ),
µ1
(µ1 + µ2 )
(µ1 + µ2 )
q21 (s) =
f (s + µ1 ) +
f (s + µ2 )
µ2
µ1
(µ1 + µ2 )2
−
f (s + µ1 + µ2 ) − 2(µ1 + µ2 )f 2 (s + µ1 + µ2 ),
µ1 µ2
q20 (s) = f (s) − [1 −
q22 (s) = (µ1 + µ2 )f 2 (s + µ1 + µ2 ),
q23 (s) = f (s + µ1 + µ2 ),
(µ1 + µ2 )2
(µ1 + µ2 )2
−
]f (s + µ1 + µ2 )
µ21
µ22
(µ1 + µ2 )2
(µ1 + µ2 )2 2
− [µ1 + µ2 −
−
]f (s + µ1 + µ2 )
µ1
µ2
(µ1 + µ2 )2
f (s + µ2 ) + (µ1 + µ2 )2 f 3 (s + µ1 + µ2 )
−
µ21
(µ1 + µ2 )2
−
f (s + µ1 )
µ22
q30 (s) = f (s) − [1 −
q31 (s) =
(µ1 + µ2 )2
(µ1 + µ2 )2
f
(s
+
µ
)
+
f (s + µ2 )
1
µ22
µ21
(µ1 + µ2 )2
(µ1 + µ2 )2
−[
+
]f (s + µ1 + µ2 )
µ21
µ22
(µ1 + µ2 )2
(µ1 + µ2 )2 2
−[
+
]f (s + µ1 + µ2 )
µ1
µ2
− 2(µ1 + µ2 )2 f 3 (s + µ1 + µ2 ),
q32 (s) = (µ1 + µ2 )2 f 3 (s + µ1 + µ2 ),
q33 (s) = (µ1 + µ2 )f 2 (s + µ1 + µ2 ),
q34 (s) = f (s + µ1 + µ2 ),
3.
Steady-state probabilities and the loss probability
The stream of overow analysis for GI/M/n/n heterogeneous-server
queueing system without waiting space is presented by Isguder and UzunogluKocer (2014) and both the steady-state probabilities and the loss probability are obtained as a function of the transition probabilities. Based on
the results given by Isguder and Uzunoglu-Kocer (2014), the steady-state
probabilities and the loss probability for GI/M/2/K queueing system are
obtained respectively as follows:
Pn =
Dnn (0)
,
n = 1, 2, ..., K.
D(1, 1, ..., 1)
p01 p12 ...pK−1,K
Ploss =
,
D(1, 1, ..., 1)
(12)
where Dnn (0) are the cofactors of the (n, n)th entries of matrix [I − q(0)]
and
1
−p01
0
···
0
−p12
···
0
1 1 − p11
.
.
.
.
.
.
.
.
.
.
D(1, 1, ..., 1) = .
.
.
.
.
1 −pK−1,1 −pK−1,2 · · · −pK−1,K 1 −p
K−1,1 −pK−1,2 · · · 1 − pK−1,K
By using (12) the loss probability for the GI/M/2/4 queueing model is
obtained as following:
p01 p12 p23 p34
,
Ploss =
(1 − p11 + p01 )A + p12 B + p01 p12 C
where, A = (1−p22 )(1−p33 −p34 )−p23 p32 , B = −p21 (1−p33 −p34 )−p23 p31 ,
C = (1 − p33 − p34 + p23 , and
1
[f (µ1 ) + f (µ2 )],
2
= f (µ1 ) + f (µ2 ) − 2f (µ1 + µ2 ),
p01 =
p11
p12 = f (µ1 + µ2 ),
p21 =
(µ1 + µ2 )
(µ1 + µ2 )
f (µ1 ) +
f (µ2 )
µ2
µ1
(µ1 + µ2 )2
−
f (µ1 + µ2 ) − 2(µ1 + µ2 )f 2 (µ1 + µ2 ),
µ1 µ2
p22 = (µ1 + µ2 )f 2 (µ1 + µ2 ),
p23 = f (µ1 + µ2 ),
p31 =
(µ1 + µ2 )2
(µ1 + µ2 )2
f
(µ
)
+
f (µ2 )
1
µ22
µ21
(µ1 + µ2 )2
(µ1 + µ2 )2
−[
+
]f (µ1 + µ2 )
µ21
µ22
(µ1 + µ2 )2 2
(µ1 + µ2 )2
−[
+
]f (µ1 + µ2 )
µ1
µ2
− 2(µ1 + µ2 )2 f 3 (µ1 + µ2 ),
p32 = (µ1 + µ2 )2 f 3 (µ1 + µ2 ),
p33 = (µ1 + µ2 )f 2 (µ1 + µ2 ),
p34 = f (µ1 + µ2 ).
4.
Conclusions
Heterogeneous server GI/M/2/K queuing system is analyzed by semiMarkov process and embedded Markov chain of the process is obtained.
The steady-state probabilities are obtained and the loss probabilities are
calculated. Since the steady state probabilities are expressed using the
determinant, these probabilities can be computed easily once the transition
matrix is known. The calculation of the average number waiting in line
and also obtaining the distribution function of the waiting time may be
the further research directions.
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2.
3.
4.
5.
6.
7.
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Gumbel M.
ÓÄÊ 004.4