UDC 519.872
∞
Study of the MAP/GI/
Queueing System with
Random Customers' Capacities
E. Lisovskaya∗ , S. Moiseeva∗
∗
Department of Probability Theory and Mathematical Statistics,
Tomsk State University,
36 Lenin ave., Tomsk, 634050, Russia
In this paper a queuing system with an innite number of servers
is considered. Customers arrive in the system according to a Markovian Arrival
Process. Each customer carries a random quantity of work (capacity of the customer). In this study service time does not depend on the customers' capacities;
the latter are used just to x some additional features of the system evolution.
It is shown that the joint probability distribution of the customers' number and
total capacities in the system is two-dimensional Gaussian under the asymptotic
condition of an innitely growing service time.
Abstract.
Innite-server queuing system, random capacity of customers, Markovian Arrival Process.
Keywords:
1. Introduction
In this work we consider an innite-server queuing system, fed by nonPoisson arrivals with random customers' capacities. Queues with random
customers' capacities are useful for analysis and design issues in highperformance computer and communication systems, in which service time
and customer volume are independent quantities [1, 2]. For instance, in [2]
performance analysis of LTE (Long Term Evolution) networks is carried
out in terms of ow-level dynamics and the amount of required radio resources does not depend on the duration of the ow. Such queues are also
important in modeling devices, where it is necessary to calculate a sucient volume of buer for data storing [3]. Results for single-server queues
with limited buer and LIFO service discipline were presented in [4], where
algorithms for the calculation of stationary characteristics were derived.
The main contribution of this paper consists in extending such analysis,
focusing on the properties of the two-dimensional process describing the
number of customers and the total capacity in the system when an inniteserver queue is fed by Markovian Arrival Process (MAP) arrivals with
random capacities and non-exponential service time distribution.
2. Matematical model
Consider a queue with innite number of servers and assume that customers arrive according to an MAP. The input process is dened by its
generator matrix
Q = ||qij || of size K ×K , the conditional rates λ1 , . . . , λK
Λ = diag{λ1 , . . . , λK } and
D = ||dij ||, where dii = 0. Denote the underlying
Markov chain of the MAP as k(t) ∈ 1, 2, . . . , K . Let each customer has
some random capacity v > 0 with distribution function G(y). An arriving
typically composed into the diagonal matrix
the probability matrix
customer instantly occupies a server in the system and its service time has
distribution function
B(x);
when the service is completed, the customer
leaves the system. Customers' capacities and service times are mutually
independent and do not dependent on the epochs of customers' arrivals.
Denote by
time
t
i(t)
and
V (t)
the number of customers in the system at
and their total capacity, respectively. Let us obtain the probabilis-
tic characteristics of two-dimensional process
{i(t), V (t)}.
This process is
not Markovian, therefore, we use the dynamic screening method for its
investigation.
We introduce a function
condition
at time
0 ≤ S(t) ≤ 1.
t,
S(t)
(dynamic probability) that satises the
We assume that a customer, arriving in the system
is screened to process with probability
with probability
Let the system be empty at moment
moment
T
in the future.
arriving at time
S(t),
and not screened
1 − S(t).
t
S(t)
t0 ,
and let us x some arbitrary
represents the probability that a customer
will be serviced in the system by the moment
S(t) = 1 − B(T − t), for t0 ≤ t ≤ T .
n(t) and W (t) the number of arrivals screened
T.
It is
easy to show [5] that
Denote by
moment
t
easy to prove the same property for the extended process
It is
{i(t), V (t)}:
P {i(T ) = m, V (T ) < z} = P {n(T ) = m, W (T ) < z}
for all
before the
in screned process and their total capacity, respectively.
(1)
m = 0, 1, 2, . . . and z ≥ 0. We use Equalities (1) for the investigation
{i(t), V (t)} via the analysis of the process {n(t), W (t)}.
of the process
3. Kolmogorov dierential equations
{k(t), n(t), W (t)}.
P (k, n, w, t) =
Let us consider the three-dimensional Markovian process
Denoting the probability distribution of this process by
P {k(t) = k, n(t) = n, W (t) < w}
and taking into account the formula of
total probability, we can write the following system of Kolmogorov dierential equations:
 w

Z
∂P (k, n, w, t)
= λk S(t)  P (k, n − 1, w − y, t)dG(y) − P (k, n, w, t) +
∂t
0
+
X
ν
qνk [P (ν, n, w, t)+
 w

Z
+dνk S(t)  P (ν, n − 1, w − y, t)dG(y) − P (ν, n, w, t)
0
for
k = 1 . . . K ; n = 0, 1, 2, . . .; w > 0.
We introduce the partial characteristic function:
h(k, u1 , u2 , t) =
∞
X
n=0
e
ju1 n
Z∞
eju2 w P (k, n, dw, t),
0
√
−1 is the imaginary unit. Then we can write the following
where j =
matrix equation:
∂h(u1 , u2 , t)
= h(u1 , u2 , t) BS(t) eju1 G∗ (u2 ) − 1 + Q
∂t
(2)
with the initial condition
h(u1 , u2 , t0 ) = r,
where
G∗ (u) =
R∞
ejuy dG(y), B = Λ + Q ⊗ D
(3)
(here
Q⊗D
is Hadamard
0
product),
r(K)]
h(u1 , u2 , t) = [h(1, u1 , u2 , t), . . . , h(K, u1 , u2 , t)], and r = [r(1), . . . ,
represents the stationary distribution of the underlying Markov
chain, i.e., vector r satises the following equations:
where e is a column vector with all entries equal to 1.
rQ = 0, re = 1,
4. Asymptotic analysis
In general, the exact solution of Equation (2) is not available, but it
may be found under asymptotic conditions. In the paper, we consider the
case of innitely growing service time.
Denoting by
Z∞
b1 =
0
Z∞
xdB(x) = (1 − B(x))dx.
0
the mean service time, then the asymptotic condition is
b1 → ∞.
We solve Problem (2)-(3) under such asymptotic condition and we obtain approximate solutions with dierent order of accuracy, named as rst-
h(u1 , u2 , t) ≈ h(1) (u1 , u2 , t)
h(u1 , u2 , t) ≈ h(2) (u1 , u2 , t).
order asymptotic
totic
We formulate the following statement.
and second-order asymp-
Lemma. The rst-order asymptotic characteristic function of the
probability distribution of the process {k(t), n(t), W (t)} has the form


Zt


h(1) (u1 , u2 , t) = r exp (ju1 κ1 + ju2 κ1 a1 ) S(v)dv ,


t0
where κ1 = rBe, and a1 =
R∞
ydG(y) is the mean customer capacity.
0
The main result of second-order asymptotic analysis is the following
theorem.
Theorem. The second-order asymptotic characteristic function of the
probability distribution of the process {k(t), n(t), W (t)} has the form
(2)
h
(u1 , u2 , t) = r exp


Zt
(ju1 κ1 + ju2 κ1 a1 )

S(v)dv+
t0


Zt
Zt
(ju1 )2 
κ1 S(v)dv + κ2 S 2 (v)dv  +
+
2
t0

(ju2 )2 
+
κ1 a2
2

Zt
t0
S(v)dv + κ2 a21
Zt
t0
t0
Zt
Zt
+ju1 ju2 κ1 a1
S(v)dv + κ2 a1
t0
where κ2 = 2g(B − κ1 I)e, a2 =
t0
R∞

S 2 (v)dv  +


S 2 (v)dv  ,

y 2 dG(y) and the row vector g satises
0
the linear matrix system
(
gQ = r(κ1 I − B),
ge = const.
Corollary. Assuming t = T and t0 → −∞ and using Equalities
, we obtain the steady-state characteristic function of the process under
study {i(t), V (t)}:
(1)
h(u1 , u2 ) = exp (ju1 κ1 b1 + ju2 κ1 a1 b1 ) +
(ju1 )2
(κ1 b1 + κ2 b2 ) +
2
(ju2 )2
2
+
κ1 a2 b1 + κ2 a1 b2 + ju1 ju2 (κ1 a1 b1 + κ2 a1 b2 ) ,
2
where
(4)
Z∞
Z∞
b1 = (1 − B(v))dv, b2 = (1 − B(v))2 dv.
0
0
From the form of the characteristic function (4), it is clear that the
probability distribution of the two-dimensional process
{i(t), V (t)} is asymp-
totically Gaussian.
5. Conclusion
In the paper, the queue with MAP arrivals, innite number of servers
and non-exponential service time is considered. Moreover, random customers' capacities, independent of their service time, are assumed. The
analysis is performed under the asymptotic condition of an innitely growing service time. It is shown that two-dimensional probability distribution
of customers' number and total capacity in the system is two-dimensional
Gaussian under this asymptotic condition.
Acknowledgments
This work is supported by Russian Foundation for Basic research,
project 16-31-00292.
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