Asymptotic Bounds for the Fluid Queue Fed by Subexponential on/off Sources

Vincent Dumas

INRIA-Rocquencourt

Algorithms Seminar

February 4, 1999

[summary by Jean-Marc Lasgouttes]

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This talk presents results from Dumas and Simonian  on the tail behaviour of the buffer content of a fluid queue processing the input of several exponential and subexponential sources. While the results in  are rather general, the presentation given here uses a simplified setting, for the sake of understandability.

## 1   Framework

Consider a fluid queue with infinite buffering capacity and outflow rate c. This queue is fed by N>1 independent stationary on/off sources, where source i, 1£ i£ N is characterized by:
• silence periods, where it generates no traffic, of length Sin, n³ 1, i.i.d. and exponentially distributed;
• activity periods, where it generates traffic at peak rate hi, of length Ain, n³ 1; these variables are i.i.d., but no assumption is made on their distribution for now.
The following notation will be useful later:
pi:=
E[Ain]
E[Ain+Sin]
,   ri := hi pi.
To characterize the stationary regime of source i, it is convenient to introduce the time elapsed in the current activity period Ai*, whose distribution is given by
Pr[Ai*=0] = 1-pi,
Pr[Ai*>x|Ai*>0]
= ó
õ
 ¥ x
Pr[Ain>y]
E[Ain]
dy.

In what follows, we restrict ourselves to the case where hiº h, piº p and riºr, for all 1£ i£ N. If Vt is the volume of fluid in the buffer at time t (with V0=0), then the following result is well known:
Theorem 1   Let Wi[t] be the flow emitted by source i in stationary regime in the interval ]-t,0] and define W[t] := åi=1NWi[t]. Then, assuming Nr<C,
 lim t®¥
Vt
 L =
V :=
 sup t³ 0
(W[t]-ct).
It is important to have good estimates for Pr[V>x], since this can be used to determine loss rate in a finite buffer queue. A typical result in this respect is due to Anick, Mitra and Sondhi : if there exist constants ai such that Pr[Ain>x] = O(e-ai x), 1£ i£ N, then there exists a such that Pr[V>x]=O(e-a x).

However, recent studies have shown that some sources may have subexponential activity patterns, such as Pr[Ain>x] =O(x-si), si>1. The purpose of this work is therefore to find good estimates for the tail distribution of V when the sources are a mix of exponential and subexponential sources, extending the results of [2, 4, 5].

## 2   Lower and Upper Bounds

Let I be a subset of {1,...,N}, with cardinal |I|, and define
AI* :=
 min iÎ I
Ai*,     W

 _ I
[t]:=
 å iÏI
Wi[t],
n0 := inf{n³0|nh+(N-n)r>c}.

Then the following bound holds as x®¥:
Pr[V>x]
³
 max I
Pr [ (|I|h+(N-|I|)r-c)AI*>x ]

³
 max |I|=n0
 Õ iÎ I
Pr [ (n0h+(N-n0)r-c)Ai*>x ] .

Similarly, defining Vi as
Vi:=
 sup t³0
( Wi[t]-r(1+e)t ) ,
where e>0 is such that (n0-1)h+(N-n0+1)r(1+e)=c, one has
Pr [V> x] £
 å |I|=n0
 Õ iÎ I
Pr é
ê
ê
ë
Vi>
x
N-n0+1
ù
ú
ú
û
.

## 3   Application to a Mix of Exponential and Subexponential Sources

Assume that the queues can be partitioned in two classes for some N0<N:
Pr[Ain>x] =O(x-s),     1£ i£ N0,
Pr[Ain>x]
=O(e
 -ai x
),
N0< i£ N.
Then the main result of this study is as follows.
Theorem 2   The following approximations hold:
• if N0<n0, then Pr[V>x]=O(e-a x);
• if N0³ n0, then Pr[V>x]=O(x-n0(s-1)).

## References


Anick (D.), Mitra (D.), and Sondhi (M. M.). -- Stochastic theory of a data-handling system with multiple sources. The Bell System Technical Journal, vol. 61, n°8, 1982, pp. 1871--1894.


Boxma (O. J.). -- Regular variation in multi-source fluid queue. In Ramaswami (V.) and Wirth (P. E.) (editors), Teletraffic Contributions for the Information Age. pp. 391--402. -- North-Holland, Washington DC, 1997. Proceedings ITC-15.


Dumas (V.) and Simonian (A.). -- Asymptotic bounds for the fluid queue fed by sub-exponential On/Off sources. -- Technical Report n°98028, MAB, Université de Bordeaux 1, 1998.


Jelenovicz (P. R.) and Lazar (A. A.). -- Asymptotic results for multiplexing on/off sources subexponential on periods. Advances in Applied Probability, vol. 31, n°2, 1999.


Rolski (Tomasz), Schlegel (Sabine), and Schmidt (Volker). -- Asymptotics of Palm-stationary buffer content distributions in fluid flow queues. Adv. in Appl. Probab., vol. 31, n°1, 1999, pp. 235--253.

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