We discuss the application of the simulation to estimate the loss probability in a queueing system with a finite buffer, which is fed by a Gaussian input. Such estimation is based on known approximation, but less known than estimation of the overflow probability in an infinite buffer system. We focus on queues with fractional Brownian input (fBi) and Brownian input (Bi). For the Bi, regenerative simulation is applied to analyze the accuracy of the analytical approximation.
This work is supported by Russian Foundation for Basic research, project No 10-07-00017

Lukashenko, O.

Morozov, E.

Pagano, M.

English

Электронная библиотека БГУ

ESTIMATION OF LOSS
PROBABILITY IN GAUSSIAN
QUEUES
O. Lukashenko
1
, E. Morozov
1
, M. Pagano
2
1
IAMR KarSC RAS,
2
University of Pisa
1
Petrozavodsk, Russia
2
Pisa, Italy
lukashenko-olegmail.ru, emorozovkarelia.ru,
m.paganoiet.unipi.it
We disuss the appliation of the simulation to estimate the loss probability in a
queueing system with a nite buer, whih is fed by a Gaussian input. Suh estimation
is based on known approximation, but less known than estimation of the overow
probability in an innite buer system. We fous on queues with frational Brownian
input (fBi) and Brownian input (Bi). For the Bi, regenerative simulation is applied to
analyze the auray of the analytial approximation.
This work is supported by Russian Foundation for Basi researh, projet No 10-
07-00017.
Keywords: uid queues; Gaussian input; loss probability
1. INTRODUCTION
Nowadays Gaussian proesses are well-reognized models to desribe the tra dy-
namis of a wide lass of modern teleommuniation networks [8℄. In this work, we use
an approximation of the loss probability expressed via overow probability for innite
buer model, whih is usually more available, to develop estimation of the required
loss probability. In partiular, we are interested in systems with small or moderate
size buers, beause the innite buer approximation is not a realisti assumption in
real networks. In this framework, the loss rate predition may represent a useful tool
to dene new all admission ontrol algorithms in Quality of Servie (Qos) supporting
network arhitetures and for QoS routing protools.
It is well-known that in the modern ommuniation networks, eah node is usually
fed by a superposition of a large number of the individual (independent) streams. More-
over, under appropriate saling suh a superposition onverges weakly to a Gaussian
proess or to an -stable Levy proess [6℄. We limit our analysis to the Gaussian ase
[8℄. More exatly, to motivate our interest to systems fed by fBi or Bi, onsider M
independent identially distributed (i.i.d) on-o soures. It then follow that, in ase
heavy-tailed on-o periods, appropriately saled umulative workload A(T t) arrived
142
during period [0; T t℄ onverges weakly to a frational Brownian motion (fBm) provided
that [10℄ rst, number of soures M ! 1, and then saling fator T ! 1. When
both on- and o-periods have nite variane, the limiting proess is standard Brownian
motion (Bm).
Denote by A(s; t) the amount of data (input tra) arrived into a ommuniation
node within time interval [s; t℄; t  s  0. Denote A(t) = A(0; t) and assume that the
input proess fA(t); t  0g has stationary inrements: the expeted workload arrived
in interval [s; t℄ is EA(s; t) = m  (t  s), where m = EA(1) is the mean input rate, and
V arA(s; t) = (t  s) for some funtion .
2. QUEUE WITH GAUSSIAN INPUT
Above given omments motivate the following denition of the Gaussian input whih
is used below:
A(t) = mt +
p
amB
H
(t); (1)
where fB
H
(t); t  0; g is a fBm, whih desribes random utuations of the input
around its linearly inreasing mean. It is assumed that queueing system has single
server with onstant servie rate C. To guarantee stability of suh a system we assume
that m < C.
The stationary overow probability in ontinuous time has the following represen-
tation [5℄:
P(Q > b) = P
 
sup
t2[0;1)
(A(t)  Ct) > b
!
: (2)
This distribution is available in an expliit form in ase of Bi. Namely,
P(Q > b) = e
( 2
C m
am
b
)
:(3)
(For more details see [9℄). In the ase of fBi there is no suh kind of exat result, but
there are some asymptotis for tail probabilities [1, 3, 7℄. Above given result relates to
an innite buer system, but our main purpose is to estimate the loss probability in a
nite buer system. Expliit or asymptoti results for nite buer systems are muh
less available (at least for non Markovian systems), and simulation is often the only
way to alulate the orresponding performane measures.
2.1. Loss probability in disrete time queues. Let B

H
(n) = B
H
(n) B
H
(n 
1); n  1. Then queue ontent in innite buer system satises (in disrete time) the
well-known Lindley-type reursion [8℄:
Q(t) =
 
Q(t  1)  C +m+
p
amB

H
(t)

+
; t = 1; 2; : : : (4)
where (x)
+
= max(0; x). Reursion (4) an be extended to the queue ontent Q
b
(t) in
a system with buer size b <1 as follows:
Q
b
(t) = min
 
(Q
b
(t  1)  C +m+
p
mB

H
(t))
+
; b

; t = 1; 2; :::: (5)
143
The time average loss P
`
(b; T ) in this system during (disrete-time) interval [0; T ℄, is
naturally dened as the ratio of the amount of loss to the total amount of input during
this interval, that is
P
`
(b; T ) =
P
T
k=1
(Q
b
(k   1) +m +
p
amB

H
(k)  C   b)
+
A(T )
: (6)
Under stability assumption, one an expet that stationary loss ratio onverges to sta-
tionary loss probability P
`
(b), that is
P
`
(b) = lim
T!1
P
`
(b; T ) =
E(Q
b
(n  1) +m +
p
amB

H
(n)  C   b)
+
m
: (7)
(In general, the existene of suh limit requires a proof beause input is long-range
dependent). The following expression given in [4℄ allows to alulate the loss probability
P
`
(b) 
P
`
(a)
P(Q > a)
P(Q > b); (8)
provided there is expliit formula (or at least a satisfatory approximation) for overow
probability P(Q > x).
The relevane of this formula is that it allows to estimate loss probability via esti-
mate of overow probability in the orresponding innite-buer system. The latter is
typially muh more easy than diret alulation of the loss probability beause there
are expliit expressions for suh a probability. Using (8), it is possible to derive estimate
for the loss probability P
`
(b):
P
`
(b)  e
 (m
b
=2)
; (9)
where
 =
1
m
p
2am
e
(
(C m)
2
2am

R
1
C
(x C)e
( 
(x m)
2
2am
!
dx;
and
m
b
:= min
n1
(b+ (C  m)n)
2
amn + 2
n 1
P
k=1
am(n  k)r(k)
;
where r(n) is the autoovariane funtion of the inrements fB

H
(n)g, whih has the
following expression:
r(n) =: ov (B

H
(0); B

H
(n)) = 1=2[(n+ 1)
2H
  2n
2H
+ jn  1j
2H
℄:
Note that for Bi we have exat value for m
b
:
m
b
= 4 
C  m
am
 b:
Estimate (9) is based on maximum variane asymptoti (MVA) approximation of the
overow probability P(Q > b) (for more details see [4℄).
144
We note that in the ase of Bi basing on expliit results (3) for ontinuous time
queue we an obtain from (8) the following approximation for the loss probability:
P
`
(b)  P
`
(0)e
( 2
C m
am
b
)
;(10)
where
P
`
(0) =
1
m
p
2am
Z
1
C
(x  C)e
( 
(x m)
2
2am

dx:
Beause Bi has independent inrements, it is possible to develop regenerative analysis of
the workload proess Q
b
(t) in the system with Bi. But stritly speaking, it is impossible
to give orret onstrution of lassial regenerations for ontinuous time Brownian
proess. This diulty is illustrated by formula (3) whih implies P(Q = 0) = 0.
Hopefully, this problem an be overome in disrete-time (slotted) setting.
For the Bi=D=1=n system we an onstrut regeneration points for the queue ontent
proess. (More details an be found in [2℄.) Let 
0
= 0 and

k+1
= minft > 
k
: Q(t  1) = 0; Q(t) > 0; k  1g; (11)
where Q(t) is the queue ontent at the end of slot t.
Now we denote by L
b
(t) the total load lost in time interval [0; t℄. Denote by EL the
mean load lost per regenerative yle and let EA be the mean load arrived per regener-
ative yle. Applying regenerative method, we obtain the following representation for
the steady-state loss probability
P
`
(b) = lim
t!1
L
b
(t)
A(t)
=
EL
EA
:
3. SIMULATION RESULTS
Now we present some numerial results related to estimation of the loss probability
for a single-server system Bi=D=1=b with onstant servie rate C and nite buer of
size b.
Figure 1 ompares the simulation results (based on the regenerative approah de-
sribed above) for the nite buer system Bi=D=1=b with the approximations (9) and
(10). The following parameters are used: C = 1; m = 0:7; N = 10
6
(where N denotes
the number of simulated time slots), the reliability of the ondene interval is 95%. It
is worth mentioning that for small buers the analytial results are outside the on-
dene interval. Moreover, as expeted from theory, (9) and (10) give upper bounds, i.
e. onservative results for dimensioning the system.
145
 1e-06
 1e-05
 0.0001
 0.001
 0.01
 0.1
 1
 0  2  4  6  8  10  12  14
Lo
ss
 P
ro
ba
bi
lity
Buffer Size
Regenerative Simulation
MVA
Continuous Time Approximation
Fig. 1. Estimate of P
`
in Bi/D/1/b: regenerative method vs. approximation (10)
4. CONCLUSIONS
In this note, we disussed the evaluation of the loss probability in a nite buer
system fed by a Gaussian input. This has beeb done using approximation (and ex-
pliit formula) for overow probability in the orresponding innite buer model. We
have ompared the so-alled MVA approximation and the estimation using regenera-
tive simulation in the ase of Brownian input. In the following researh we also plan
to simulate the queue (using standard Monte Carlo algorithms) with general fBi and
ompare estimation of the overow or loss probability with the approximation (9).
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147


Estimation Of Loss Probability In Gaussian Queues

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Estimation Of Loss Probability In Gaussian Queues

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