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

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 [3] 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 [3] 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 S_in, n³ 1, i.i.d. and exponentially distributed;
activity periods, where it generates traffic at peak rate h_i, of length A_in, 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:

p_i:=

E[A_in]

E[A_in+S_in]

, r_i := h_i p_i.

To characterize the stationary regime of source i, it is convenient to introduce the time elapsed in the current activity period A_i^*, whose distribution is given by

Pr[A_i^*=0]

= 1-p_i,

Pr[A_i^*>x|A_i^*>0]

ó
õ

Pr[A_in>y]

E[A_in]

dy.

In what follows, we restrict ourselves to the case where h_iº h, p_iº p and r_iºr, for all 1£ i£ N. If V_t is the volume of fluid in the buffer at time t (with V₀=0), then the following result is well known:

Theorem 1 Let W_i[t] be the flow emitted by source i in stationary regime in the interval ]-t,0] and define W[t] := å_i=1^NW_i[t]. Then, assuming Nr<C,

lim

t®¥

V_t

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 [1]: if there exist constants a_i such that Pr[A_in>x] = O(e^-a_ⁱ^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[A_in>x] =O(x^-s_ⁱ), s_i>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

A_I^* :=

min

iÎ I

A_i^*, W

[t]:=

iÏI

W_i[t],

n₀ := inf{n³0|nh+(N-n)r>c}.

Then the following bound holds as x®¥:

Pr[V>x]

max

[

(|I|h+(N-|I|)r-c)A_I^*>x

]

max

|I|=n₀

iÎ I

[

(n₀h+(N-n₀)r-c)A_i^*>x

]

Similarly, defining V_i as

V_i:=

sup

t³0

(

W_i[t]-r(1+e)t

)

where e>0 is such that (n₀-1)h+(N-n₀+1)r(1+e)=c, one has

[V> x] £

|I|=n₀

iÎ I

é
ê
ê
ë

V_i>

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[A_in>x]

=O(x^-s),

1£ i£ N₀,

Pr[A_in>x]

=O(e

-a_i 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^-n_⁰^(s-1)).

References

[1]: 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.
[2]: 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.
[3]: 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.
[4]: 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.
[5]: 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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