STOCHASTIC SERVER MODEL
WITH LIMITED STORAGE SIZE
Marcin Ziółkowski
Jan Długosz University
Częstochowa, Poland
E-mail: marionesta5@wp.pl
In this paper analytic model of server servicing requests of many users is presented.
Every user can dispose of computer system having its own characteristics and can be
represented by classical
queue but in fact users are dependent because they are
connected via common memory space which is limited. Analogous models can represent
mail or ftp servers and can be used to calculate needed server storage size ([1, 2]). In this
research stationary distribution function of number of requests and stationary refusal
probability for every user was obtained. It contains also review of some special cases.
Keywords: Markovian process;
inversion; Stieltjes convolution.
queue; Loss probability; Laplace transform
1. MODEL DESCRIPTION AND ANALYSIS
In the beginning combination of two
computer systems is investigated.
For such one number of requests distribution function and refusal probability for every
system will be obtained - both in the steady state. Let us then consider two classical
independent
systems denoting them as
. Let
be
suitable entrance Poisson flow and service time parameters for i-th queueing system. In
addition we assume that every request in i-th system has some random size . Let
be
distribution function of this random variable. Denote as
- summary size of all requests
present in server storage at time t and assume that server storage size is limited by value V
and requests service times and their sizes are independent. Let
be the number of
requests present in i-th system at time t,
- the size of j-th request in i-th system at
time t,
. Then the combination of two systems can be described by the
following Markovian process
(1)
Process (1) can be characterized by the following functions
(2)
(3)
(4)
It is clear that steady state exists for analysed combination if value of V is finite. Then
1
we can define functions
if
and describe random variables
the sense of weak convergence.
and that are limits of functions (2)-(4)
that are the limits of processes
in
Now we will obtain stationary number of requests distribution function.
Let
be Cronecker symbol :
. We will also use the following
notations:
following equations for functions
. Analysing process (1) we can write the
and
:
;
(5)
;
(6)
;
(7)
+
,
.
(8)
Denote as
the k-th order Stieltjes convolution of
functions, and for
convolution of distribution functions
of non-negative random variables
we will use the following notation
.
We introduce also the notation
2
(9)
By the substitution we can check that solution of the (5)-(8) system has the form
(10)
From (10) we easily obtain
(11)
The value of
can be obtained from normalization condition
(12)
Obtained results can be extended for the arbitrary r number of computer systems. Then we
have
The value of
can be obtained from analogous normalization condition.
2. REFUSAL PROBABILITY
Stationary refusal probability for i-th system (i=1,2) can be obtained from the
following equilibrium condition (in the steady state, during the same interval of time, the
mean number of arriving requests that are not lost is equal to the mean number of serviced
requests).
where
.
From (13) we obtain the following formula
Analogous formula can be obtained for the arbitrary number of systems.
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3. RESULTS IN SOME SPECIAL CASES
Now we will investigate some special case - combination of two one-device systems
connected via common memory space. Generalization for the arbitrary number of systems
is obvious. In practice this case is the model of server servicing requests of some systems
consisting of single computer. In addition we suppose that all requests have the same size
distribution function which is exponential with the same parameter f and
If
for i=1,2 and
then we obtain
(15)
where
If
.
then we obtain
(16)
where
If
.
then we obtain
,
(17)
where
If
then we have
,
(18)
where
Obtained formulas can be extended for the arbitrary number of systems.
During the analysis, calculating loss probabilities, we can notice that investigated
combination of one-device systems with exponential request distribution function with the
same parameter f for every system can belong to one of the following classes:
-
stable state (
overload state (
for every i) – then
for at least one i) – then
;
.
If the parameters of exponential request distribution function are not the same then refusal
probabilities vary.
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The simulation results presented in table 1 show that fact.
Table 1. Refusal probabilities for
V
1
2
3
4
5
6
7
8
0,600
0,454
0,368
0,310
0,267
0,236
0,211
0,191
,
0,400
0,273
0,210
0,172
0,146
0,128
0,112
0,101
Now we will investigate one more special case. Let us suppose that we have combination of
two one-device systems with finite number of waiting places. In addition we assume that
request distribution function is uniform on the [a,b] interval and number of waiting places
is the same for both systems.
Using Laplace transform inversion we can find formula for the k-th order Stieltjes
convolution of uniform distribution functions on the [a,b] interval which has the form
where H(x) is a Heaviside unitstep function.
Then from (11) and (14), using (19), we can calculate numerical results for
probabilities.
We can notice (after calculations) that in this case refusal probabilities
are not the
same and converge to 0 in stable state or to the
in the overload state.
LITERATURE
1. Tikhonenko, O.M.: Queueing systems of a random length demands with restrictions. Automation and
Remote Control 52 (10, pt. 2), 1431--1437 (1991)
2. Tikhonenko, O.: Probability Analysis of Information Systems. EXIT. Warsaw (2006)
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