Asymptotics of Palm-Stationary Buffer Content Distributions in Fluid Flow Queues

Tomasz Rolski^{∗}  Sabine Schlegel^{†}  Volker Schmidt^{‡}

December 19, 1997

Abstract

We study a fluid flow queueing system with m independent sources alternating between silence and activity periods; m ≥ 2. The distribution function of the activity periods of one source, is supposed to be intermediate regular varying. We show that the distribution of the net increment of the buffer during an aggregate activity period (i.e. at least one source is active) is asymptotically tail-equivalent to the distribution of the net input during a single activity period with intermediate regular varying distribution function. In this way, we arrive at an asymptotical representation of the Palm-stationary tail-function of the buffer content at the beginning of aggregate activity periods. Our approach is probabilistic and extends recent results of Boxma (1996, 1997) who considered the special case of regular variation.

Keywords: Regular variation, Subexponential distribution, Aggregating, Tail asymptotics, Random walk, On-off fluid flow

AMS 1991 Subject Classification: 60K25; 60F10; 68M20; 90B15.

∗Postal address: Mathematical Institute, Wroclaw University, pl. Grunwaldzki 2/4, 50–384 Wroclaw, Poland

†Postal address: Institute of Stochastics, University of Ulm, Helmholtzstraße 18, D-89069 Ulm, Germany

‡Postal address: Institute of Stochastics, University of Ulm, Helmholtzstraße 18, D-89069 Ulm, Germany

1 Introduction

Motivated by data flows in telecommunication networks, there is an increasing interest in fluid flow queues with merging on-off sources having heavytailed governing distributions, see Anantharam (1995), Boxma (1996, 1997), Choudhury and Whitt (1997), Heath, Resnick and Samorodnitsky (1996a,b), Heyman and Lakshman (1996), Jelenković and Lazar (1996), Whitt (1993). The present paper is devoted to the study of a fluid flow queueing system with m independent sources alternating between silence and activity periods; m ≥ 2. The distribution function of the activity periods of one source, say number 1, is supposed to be intermediate regular varying, where we say that a distribution function F is intermediate regular varying (F ∈ IR), if

lim_{c↓1} lim inf_{x→∞} \frac{\overline{F} (cx)}{\overline{F} (x)} = lim_{c↓1} lim sup_{x→∞} \frac{\overline{F} (cx)}{\overline{F} (x)} = 1.  (1)

We call this activity period a heavy-tailed activity period. The activitiy periods of the other sources and all silence periods are assumed to be exponentially distributed. The input rates of all sources are assumed to be greater than the output rate.

We show that the distribution of the net increment of the buffer during an aggregate activity period (i.e. at least one source is active) is asymptotically tail-equivalent to the distribution of the net input during a single heavytailed activity period. In this way, we receive an asymptotical representation of the Palm-stationary tail-function of the buffer content at the beginning of aggregate activity periods. Using Laplace-Stieltjes transforms, the same result has been obtained in Boxma (1996) for the case of two sources and extended in Boxma (1997) to m ≥ 2 sources, one of them with regular varying distribution function of activity periods which is a stronger condition than (1). In return, Boxma (1997) considers more general distributions for the activity periods of sources 2, …, m, assuming that they have an exponentially bounded tail. However, in contrast to Boxma’s proof, our approach is probabilistic and thus offers much more insight. Moreover, we conjecture that future research might show that our approach can be used for an even broader class of heavy-tailed distributions than that defined by (1).

We also remark that in Theorem 10 of Jelenković and Lazar (1996) the asymptotical tail behaviour of the time-stationary buffer content distribution has been studied for the considered model. However, it seems to be impossible to obtain the asymptotical tail behaviour of the Palm-stationary buffer content distribution directly from their result. Besides this we propose other proving techniques.

2 Model Description and Notation

We consider a fluid flow queueing system with infinite buffer size and constant unit output rate. The system is fed by m independent on-off sources, each of which has mutually independent alternating activity and silence periods. They are also called on and off periods respectively. Activity periods of source 1 are denoted by T_{n} and supposed to have an intermediate regular varying distribution F with finite mean IET, whereas activity periods of source k ≥ 2, denoted by J_{n}^{(k)}, as well as the silence periods of source 1, S_{n}, and source k ≥ 2, I_{n}^{(k)}, n = 1, 2, … have an exponential distribution with parameter λ_{on}^{(k)}, ν and λ_{off}^{(k)} respectively. The generic random variables of I_{n}^{(k)}, J_{n}^{(k)}, S_{n}, T_{n} are denoted by I^{(k)}, J^{(k)}, S, T respectively and they are assumed to be mutually independent. When active, the sources transmit at constant rate r_{k} with

r_{k} ≥ 1,  k = 1, …, m .  (2)

Furthermore, we assume that the following stability condition holds:

L =^{def} \frac{r_{1}IET ν}{1 + IET ν} + ∑_{k=2}^{m} \frac{r_{k}λ_{off}^{(k)}}{λ_{on}^{(k)} + λ_{off}^{(k)}} < 1 ,  (3)

i.e. the expected total input into the system per unit time is less than the output rate. We also assume that the buffer is empty and all sources are off at time zero, that is we start with a silence period of the whole system. By r_{max} = ∑_{k=1}^{m} r_{k} − 1 we denote the maximum net input rate.

We further use the following notation. Let C_{n}, n = 1, 2, …, be the length of the nth aggregate activity period after zero followed by a silence period whose generic random variable is defined by min{S, I^{(2)}, …, I^{(m)}}. This silence period of the whole system is exponentially distributed with parameter Λ =^{def} (ν +∑_{k=2}^{m} λ_{off}^{(k)}). By B_{n} and W_{n}, n = 1, 2, …, we denote the net increment of the buffer content during C_{n} and the buffer content at the beginning of the nth aggregate activity period respectively. Figure 1 shows a typical sample path of the on-off processes in the case of two sources as well as a typical sample path of the resulting buffer content process. Note that B_{1}, B_{2}, … are i.i.d. random variables with distribution, say, B(·) and

[figure: T_{1}; T_{2}; T_{3}; S_{1}; S_{2}; S_{3}; I_{1}^{(2)}; I_{2}^{(2)}; I_{3}^{(2)}; I_{4}^{(2)}; I_{5}^{(2)}; J_{1}^{(2)}; J_{2}^{(2)}; J_{3}^{(2)}; J_{4}^{(2)}; J_{5}^{(2)}; B_{1}; W_{2}; C_{1}; C_{2}; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .]

Figure 1: Aggregate Activity Period of Two Merging On-Off-Sources

that W_{n}, B_{n} and the following silence period of the system are independent, too. Thus we can reinterpret the development of the buffer content after successive aggregate activity and silence periods of the system as a random walk with generic increment B − min{S, I^{(2)}, …, I^{(m)}}, and W_{n} then corresponds to the finite-horizon maximum of this random walk up to step n, n = 1, 2, . . .. The distribution of the infinite-horizon maximum of this random walk we denote by W (·), i.e. W (x) = lim_{n→∞} IP(W_{n} ≤ x).

Under stability condition (3) this random walk has negative drift. The limit distribution W (·) is therefore well-defined and can be interpreted as Palm-stationary distribution of the buffer content at the beginning of aggregate activity periods. It turns out that W (·) is asymptotically tail-equivalent to the so-called tail-integrated distribution of B(·). Furthermore, B(·) is asymptotically tail-equivalent to the distribution of the net input during a single activity period of source 1. Thus, in Section 3 we begin with the investigation of the buffer content process during such a single activity period. For simplicity of notation, in Section 4 we first determine the asymptotical tail behaviour of B(·) for the case of two merging sources, i.e. m = 2. The investigation of the general case is postponed to Section 5. If m = 2, we write I_{n}, J_{n}, λ_{off}, λ_{on} instead of I_{n}^{(k)}, J_{n}^{(k)}, λ_{off}^{(k)}, λ_{on}^{(k)} respectively.

3 Net Input during a Single Activity Period of Source 1

In this section we prove two asymptotical properties of the net input during a single activity period of source 1 which are crucial for showing that the distribution B(·) asymptotically behaves like the distribution of the net input during a single activity period of source 1. However, we first state two auxiliary results for homogeneous Markov processes with finite state space.

3.1 Some Preliminaries

Let {ξ(t), t ≥ 0} be a homogeneous, irreducible Markov process with finite state space E = {0, 1, …, ℓ}. By α = (α_{0}, …, α_{ℓ}) we denote the initial distribution of {ξ(t)}, by P (t) = (p_{ij}(t)) its matrix transition function, and by Q the matrix of transition intensities. Let π = (π_{0}, …, π_{ℓ}) be the stationary (limit) distribution of {ξ(t)}, i.e. πQ = 0 where 0 = (0, …, 0), and let a = (a_{0}, …, a_{ℓ}) be an arbitrary (ℓ + 1)-dimensional vector with real-valued components such that ∑_{i=0}^{ℓ} π_{i}a_{i} < 0.

Lemma 3.1 There exist positive constants c, γ < ∞ such that for all x ≥ 0,

IP (sup_{t≥0} ∫_{0}^{t} a_{ξ(s)} ds > x) ≤ ce^{−γx} .  (4)

Lemma 3.2 There exist positive constants t_{0}, δ < ∞ such that for all t ≥ t_{0},

max_{i,j∈E} |p_{ij}(t) − π_{j}| ≤ e^{−δt} .  (5)

For a proof of Lemma 3.1 we refer to Palmowski and Rolski (1996). A proof of Lemma 3.2 can be found e.g. in Iosifescu (1980).

3.2 Basic Lemma

We apply Lemmas 3.1 and 3.2 to the following case. The random variable

r_{1} − 1 + ∑_{k=2}^{m} r_{k}1I(source k is active at time t) ,  (6)

where 1I(A) denotes the indicator of the set A, describes the net input rate at time t provided that t belongs to an activity period of source 1. The generic representation of the resulting stochastic process during an activity period of source 1 is as follows. Consider the family E of all subsets of {2, …, m}. Sometimes it is convenient to identify E with {0, 1, …, ℓ}, where we relate the empty set to 0 and clearly ℓ = 2^{m−1} − 1. Let ξ(t) be the random set of those sources with numbers from 2 to m which are active at time t. Under the assumptions of Section 2, {ξ(t), t ≥ 0} is a Markov process of the type considered in Section 3.1. Note that there and also in the present section {ξ(t)} can have an arbitrary initial distribution α. Let T be a generic activity period of source 1 and define the process {η(t), t ≥ 0} by

η(t) = r_{1} − 1 + ∑_{k=2}^{m} r_{k}1I(k ∈ ξ(t)) .  (7)

Then, supposing that the activity period of source 1 begins at t = 0, the process {η(t)} describes the net input rate in time interval [0, T]. Furthermore, define the mean rate during activity period of source 1 by μ = lim_{t→∞} IEη(t) and note that (2) implies μ > 0. But it is evident that the converse implication is not true. However, in Sections 3.2 and 3.3 we only need that μ > 0.

Let η_{T} = ∫_{0}^{T} η(s)ds. The following result is an important tool for getting the other asymptotical equivalence properties derived in the present paper.

Lemma 3.3 For x → ∞,

IP(η_{T} > x) ∼ IP(μT > x) .  (8)

Proof Let ε > 0 be fixed. Then,

IP(η_{T} > x) = IP(T > \frac{x}{r_{max}} , η_{T} > x) = φ_{1}^{+}(ε, x) + φ_{2}^{+}(ε, x) ,

where φ_{1}^{+}(ε, x) = IP(r_{max}T > x, η_{T} > x, η_{T} − μT > εT) and φ_{2}^{+}(ε, x) = IP(r_{max}T > x, η_{T} > x, η_{T} − μT ≤ εT). By Lemma 3.1 we receive

φ_{1}^{+}(ε, x) ≤ IP (T > \frac{x}{r_{max}} , \frac{1}{T} ∫_{0}^{T} (η(s) − (μ + \frac{ε}{2})) ds > \frac{ε}{2})

≤ IP (sup_{t≥0} ∫_{0}^{t} (η(s) − (μ + \frac{ε}{2})) ds > x\frac{ε}{2r_{max}} ) ≤ ce^{−γ̃x} ,

where γ̃ = γε/(2r_{max}) and c and γ according to Lemma 3.1. For the term φ_{2}^{+}(ε, x) we obtain the following upper bound

φ_{2}^{+}(ε, x) ≤ IP(η_{T} > x, η_{T} ≤ T (μ + ε)) ≤ IP(T (μ + ε) > x) .

Combining this, we obtain

lim sup_{x→∞} \frac{IP (η_{T} > x)}{IP(μT > x)} ≤ lim sup_{x→∞} \frac{ce^{−γ̃x}}{IP(μT > x)} + lim sup_{x→∞} \frac{IP(T (μ + ε) > x)}{IP(μT > x)} .

Using that the distribution F_{T} of T is from IR, letting ε ↓ 0 finally yields

lim sup_{x→∞} \frac{IP (η_{T} > x)}{IP(μT > x)} ≤ 1 .

To get a lower bound, note that

IP(η_{T} > x) ≥ IP(η_{T} > x, η_{T} ≥ T (μ − ε))
≥ IP(T (μ − ε) > x, ∫_{0}^{T} (η(s) − (μ − \frac{ε}{2})) ds ≥ −T \frac{ε}{2})
≥ φ_{1}^{−}(ε, x) − φ_{2}^{−}(ε, x) ,

where φ_{1}^{−}(ε, x) = IP(T (μ − ε) > x) and

φ_{2}^{−}(ε, x) = IP(T (μ − ε) > x, ∫_{0}^{T} (η(s) − (μ − \frac{ε}{2})) ds < −x\frac{ε}{2(μ − ε)}) .

Application of Lemma 3.1 to the term φ_{2}^{−}(ε, x) yields

φ_{2}^{−}(ε, x) ≤ IP( sup_{t≥0} ∫_{0}^{t} ((μ − \frac{ε}{2}) − η(s)) ds > \frac{xε}{2(μ − ε)}) ≤ ce^{−γ^{∗}x} ,

where γ^{∗} = γε/(2(μ − ε)) and c and γ are from Lemma 3.1, i.e.

lim inf_{x→∞} \frac{IP (η_{T} > x)}{IP(μT > x)} ≥ lim inf_{x→∞} (\frac{IP(T (μ − ε) > x)}{IP(μT > x)} − \frac{ce^{−γ^{∗}x}}{IP(μT > x)}) ,

and thus, since F_{T} ∈ IR, ε ↓ 0 yields

lim inf_{x→∞} \frac{IP (∫_{0}^{T} η(s) ds > x)}{IP(μT > x)} ≥ 1 . □

Remark Recall that the distribution function F of the generic activity period T of source 1 is assumed to be intermediate regular varying, i.e. condition (1) is satisfied, which is slightly more general than regular variation. For details and examples we refer to Cline (1994) and Bingham, Goldie and Teugels (1987). We also mention that in the proof of Lemma 3.3 we did not use that T is independent of {ξ(t)}. Furthermore, we did not use that {ξ(t)} is Markov. Thus each process {ξ(t)} satisfying (4) and (5) is possible.

3.3 Refined Asymptotics

In Sections 4 and 5 we will use the following refinement of Lemma 3.3.

Lemma 3.4 For all i ∈ {0, 1, …, ℓ},

IP (η_{T} > x, ξ(T) = i) ∼ π_{i}IP(μT > x)  (9)

as x → ∞, where π = (π_{0}, …, π_{ℓ}) is the stationary distribution of {ξ(t)}.

Proof We subdivide the proof into two steps showing that

IP(μT > x, ξ(T) = i) ∼ π_{i}IP(μT > x) , x → ∞  (10)

and

IP(η_{T} > x, ξ(T) = i) ∼ IP (μT > x, ξ(T) = i) , x → ∞ .  (11)

In order to prove (10) we first derive an upper bound. Recall that α = (α_{0}, …, α_{ℓ}) denotes the initial distribution of {ξ(t)} and P (t) = (p_{ij}(t)) its matrix transition function. Then, using the fact that T is independent of {ξ(t)}, Lemma 3.2 gives for some δ > 0 and for all x sufficiently large,

IP(μT > x, ξ(T) = i) = ∑_{k=0}^{m} ∫_{x/μ}^{∞} (p_{ki}(s) − π_{i} + π_{i}) dF (s)α_{k}

≤ ∑_{k=0}^{m}(∫_{x/μ}^{∞} (|p_{ki}(s) − π_{i}| + π_{i}) dF (s))α_{k} ≤ (e^{−δ \frac{x}{μ}} + π_{i})\overline{F}_{T} (\frac{x}{μ}) ,

and thus

lim sup_{x→∞} \frac{IP(μT > x, ξ(T) = i)}{IP(μT > x)} ≤ π_{i} .  (12)

To get a lower bound note that

IP(μT > x, ξ(T) = i) = ∑_{k=0}^{m} ∫_{x/μ}^{∞} (p_{ki}(s) − π_{i} + π_{i}) dF (s)α_{k}

≥ ∑_{k=0}^{m}(− ∫_{x/μ}^{∞} e^{−δs}dF (s) + π_{i}\overline{F}_{T} (\frac{x}{μ}))α_{k} ≥ −e^{−δ \frac{x}{μ}} \overline{F}_{T} (\frac{x}{μ}) + π_{i}\overline{F}_{T} (\frac{x}{μ})

for x sufficiently large, and thus

lim inf_{x→∞} \frac{IP(μT > x, ξ(T) = i)}{IP(μT > x)} ≥ lim inf_{x→∞} (−e^{−δ \frac{x}{μ}} + π_{i}) = π_{i} .  (13)

Combining (12) and (13) yields (10). In order to show (11) we again derive an upper and a lower bound. Let ε > 0 be fixed. Then,

IP(η_{T} > x, ξ(T) = i) = IP(T > \frac{x}{r_{max}} , η_{T} > x, ξ(T) = i) = φ_{1}^{+}(ε, x)+φ_{2}^{+}(ε, x)

with φ_{1}^{+}(ε, x) = IP(r_{max}T > x, η_{T} > x, η_{T} − μT > εT, ξ(T) = i) and φ_{2}^{+}(ε, x) = IP(r_{max}T > x, η_{T} > x, η_{T} − μT ≤ εT, ξ(T) = i). The term φ_{1}^{+}(ε, x) can be bounded as follows:

φ_{1}^{+}(ε, x) ≤ IP(T > \frac{x}{r_{max}} , \frac{1}{T} ∫_{0}^{T} (η(s) − (μ + \frac{ε}{2})) ds > \frac{ε}{2})
≤ IP(sup_{t≥0} ∫_{0}^{t}(η(s) − (μ + \frac{ε}{2})) ds > x\frac{ε}{2r_{max}} ) ≤ ce^{−γx} ,

where c and γ > 0 are chosen according to Lemma 3.1. Furthermore,

φ_{2}^{+}(ε, x) ≤ IP(η_{T} > x, η_{T} ≤ T (μ+ε), ξ(T) = i) ≤ IP(T (μ+ε) > x, ξ(T) = i).

Note that by (10) and the fact that F ∈ IR,

lim_{ε↓0} lim sup_{x→∞} \frac{IP(T (μ + ε) > x, ξ(T) = i)}{IP(μT > x, ξ(T) = i)}
= lim_{ε↓0} lim sup_{x→∞} \frac{IP(T (μ + ε) > x, ξ(T) = i)}{IP(T (μ + ε) > x)} \frac{IP((μ + ε)T > x)}{IP(μT > x)}

×\frac{IP(μT > x)}{IP(μT > x, ξ(T) = i)} = 1

and therefore

lim sup_{x→∞} \begin{matrix}IP & (η_{T} > x, ξ(T) = i) \\ IP & (μT > x, ξ(T) = i) \end{matrix} ≤ lim sup_{x→∞} \frac{ce^{−γx}}{IP(μT > x)} \frac{IP(μT > x)}{IP(μT > x, ξ(T) = i)}

+ lim_{ε↓0} lim sup_{x→∞} \frac{IP(T (μ + ε) > x, ξ(T) = i)}{IP(μT > x, ξ(T) = i)} ≤ 1 .

On the other hand,

IP(η_{T} > x, ξ(T) = i) ≥ IP(η_{T} > x, η_{T} ≥ T (μ − ε), ξ(T) = i)
≥ IP(T (μ − ε) > x, ∫_{0}^{T} (η(s) − (μ − \frac{ε}{2})) ds ≥ −T \frac{ε}{2}, ξ(T) = i)
= φ_{1}^{−}(ε, x) − φ_{2}^{−}(ε, x) ,

where φ_{1}^{−}(ε, x) = IP(T (μ − ε) > x, ξ(T) = i) and

φ_{2}^{−}(ε, x) = IP(T (μ − ε) > x, ξ(T) = i, ∫_{0}^{T} (η(s) − (μ − \frac{ε}{2})) ds < \frac{−xε}{2(μ − ε)}) .

The term φ_{2}^{−}(ε, x) is exponentially bounded since by Lemma 3.1

φ_{2}^{−}(ε, x) ≤ IP(∫_{0}^{T} ((μ − \frac{ε}{2}) − η(s)) ds > x\frac{ε}{2(μ − ε)}) ≤ ce^{−γx}

for some c, γ > 0. Similar to the considerations on φ_{2}^{+}(ε, x) above,

lim_{ε↓0} lim inf_{x→∞} \frac{φ_{1}^{−}(ε, x)}{IP(μT > x, ξ(T) = i)} = 1 ,

and hence

lim inf_{x→∞} \frac{IP(η_{T} > x, ξ(T) = i)}{IP(μT > x, ξ(T) = i)} ≥ lim inf_{x→∞} \frac{IP(T (μ − ε) > x, ξ(T) = i)}{IP(μT > x, ξ(T) = i)}

− lim sup_{x→∞} \frac{ce^{−γx}}{IP(μT > x, ξ(T) = i)} ⟶^{ε↓0} 1 . □

4 Net Input during Aggregate Activity Periods

Consider the Markov process {ξ(t)} introduced in Section 3.2 and note that {ξ(t)} can be represented in the following way. Suppose that the initial state ξ(0) of {ξ(t)} at time t = 0 is deterministic and let {ζ_{k}^{i_{k}}(t)} be the process which takes value 1 if source k is active and 0 otherwise. The superscript i_{k} = 0, 1 indicates, whether at t = 0 the source is active (i_{k} = 1) or silent (i_{k} = 0). Then, in addition to the set representation of ξ(t) given in Section 3.2, we can write ξ(t) = (ζ_{2}^{i_{2}}(t), …, ζ_{m}^{i_{m}}(t)). Under our assumptions on the model, the processes {ζ_{k}^{i_{k}}(t)} (k = 2, …, m) are Markovian and mutually independent. Let i = (i_{2}, …, i_{m}) and in particular, 1 = (1, …, 1). We now append the superscript i to ξ(t), writing ξi(t) to emphazise that the initial state of the defining process {ζ_{k}^{i_{k}}(t)} is i_{k}; k = 2, …, m. Similarly we append the superscript i to the process {η(t)} writing ηi(t).

In the rest of the paper {ξi_{n}(t), t ≥ 0} are independent copies of {ξi(t), t ≥ 0} and I_{0}^{(k)}, I_{1}^{(k)}, … are i.i.d. exponentially distributed with parameter λ_{off}^{(k)}, J_{0}^{(k)}, J_{1}^{(k)}, … are i.i.d. exponentially distributed with parameter λ_{on}^{(k)}, S_{0}, S_{1}, … are i.i.d. exponentially distributed with parameter ν. Besides we assume that {I_{n}^{(k)}}, {J_{n}^{(k)}}, {S_{n}}, {ξi_{n}(t)} (i ∈ E) are mutually independent. Note that from now on we occasionally use a somewhat different numbering of activity and silence periods which does not always coincide with the numbering introduced at the beginning of the paper, i.e. in Section 2. Furthermore, by {ηi_{1}(t)}, {ηi_{2}(t)}, … we denote independent and identically distributed copies of the process {ηi(t)}.

To make our consideration more transparent we first consider the case m = 2. We assume that an aggregate activity period of the whole system begins at time t = 0. Now, E = {0, 1} and the random variable ξ(t) ∈ {0, 1} indicates whether source 2 is off or on at time t. Recall that π = (π_{0}, π_{1}) denotes the stationary distribution of {ξ(t)}, and hence μ = r_{1}−1+π_{1}r_{2} > 0. Then we have the following asymptotical representation of the distribution of the generic net inflow B during this aggregate activity period.

Theorem 4.1 For x → ∞,

IP(B > x) ∼ \frac{ν(λ_{on} + λ_{off})}{(ν + λ_{off})λ_{on}}IP(μT > x) .  (14)

Remark We will see in Section 4.5 that (14) can be rewritten in the form

IP(B > x) ∼ (IP(S < I) + IP(S > I)IP(S < J)) ∑_{n=1}^{∞}(1−p)p^{n−1}\overline{G^{∗n}}(x) , (15)

where G(x) = IP(μT ≤ x) and p = π_{1}ν(ν + λ_{on})^{−1}. Intuitively this asymptotical representation formula is clear. As the silence periods of source 1 are light-tailed and thus can asymptotically be neglected, the net inflow into the buffer during an aggregate activity period is governed by the inflow during activity periods of source 1 and therefore depends on the number of these activity periods. To have exactly n such activity periods each of the n − 1 silence periods inbetween must be bridged by an activity period of source 2 which is the case if at the end of an activity period of source 1 source 2 is active with the remaining time of this activity period lasting longer than the silence period of source 1. But, in steady state, this event has probability p and thus (15) is plausible.

4.1 A Representation Formula

The formal proof of Theorem 4.1 will be partitioned into steps. We first discuss some structural details and derive some generic asymptotic representation formulas. However, all these considerations are motivated by the fact that the random variable B can be represented in the following way.

Lemma 4.1

B = 1I(S_{0} > I_{0})1I(S_{1} > J_{0})(r_{2} − 1)J_{0}

+ 1I(S_{0} > I_{0})1I(S_{1} < J_{0})( ∑_{n=1}^{N^{1}} ((r_{2} − 1)S_{n} + Y_{n}^{1})
+(r_{2} − 1)J_{N^{1}}1I(ξ_{N^{1}}^{1}(T_{N^{1}}) = 1))  (16)
+ 1I(S_{0} < I_{0}){Y_{1}^{0} + (r_{2} − 1)J_{1}1I(ξ_{1}^{0}(T_{1}) = 1)1I(S_{2} > J_{1})
+ 1I(N^{0} ≥ 2)( ∑_{n=2}^{N^{0}} ((r_{2} − 1)S_{n} + Y_{n}^{1}) + (r_{2} − 1)J_{N^{0}}1I(ξ_{N^{0}}^{1}(T_{N^{0}}) = 1))} ,

where

Y_{n}^{i} = ∫_{0}^{T_{n}} η_{n}^{i}(s) ds for n ≥ 1 , η_{n}^{i}(s) = r_{1} − 1 + r_{2}1I(ξ_{n}^{i}(s) = 1) ,
N^{0} = \{ \begin{matrix}1 & ξ_{1}^{0}(T_{1}) = 0 or S_{2} > J_{1}, \\ min \{n ≥ 2 : ξ_{n}^{1}(T_{n}) = 0 or S_{n+1} > J_{n}\} & otherwise, \end{matrix}
N^{1} = min {n ≥ 1 : ξ_{n}^{1}(T_{n}) = 0 or S_{n+1} > J_{n}} .

The proof of Lemma 4.1 is omitted since it is evident.

We will use (16) in order to determine the asymptotical tail behaviour of B. In connection with this, it is useful to consider a class of distributions of nonnegative random variables which is larger than IR and which contains, besides regular varying distributions, for example also the lognormal and Weibull distributions.

4.2 Subexponential Distributions

Let S denote the class of all subexponential distributions on IR_{+} where a distribution function F is said to be subexponential if

lim_{x→∞} \frac{1 − F^{∗2}(x)}{1 − F (x)} = 2.  (17)

Lemma 4.2 Each distribution function from IR belongs to S.

Proof Using the defining equations (1) and (17), the statement of this lemma can easily be concluded from Corollary 1.2 of Cline (1994). □

Knowing that IR ⊂ S, the following properties of subexponential distributions are useful to prove certain tail behaviours of distributions from class IR.

Lemma 4.3 Let F, G be two distributions on IR_{+} and assume that there exists a constant c ∈ (0, ∞) with \overline{G}(x) ∼ c\overline{F} (x). Then, F ∈ S if and only if G ∈ S.

Lemma 4.4 Let F, G, H be distributions on IR_{+} such that F ∈ S, \overline{G}(x) ∼ c_{1}\overline{F} (x) and \overline{H}(x) ∼ c_{2}\overline{F} (x), where c_{i} ∈ (0, ∞) for i = 1, 2. Then, \overline{G ∗ H}(x) ∼ (c_{1} + c_{2})\overline{F} (x) and in particular \overline{F ∗ H}(x) ∼ (1 + c_{2})\overline{F} (x).

Lemma 4.5 Let F ∈ S and let G be a distributions on IR such that \overline{G}(x) = o(\overline{F} (x)) as x → ∞. Then, \overline{F ∗ G}(x) ∼ \overline{F} (x).

Lemma 4.6 If F ∈ S, then, for each ε > 0, there exists a constant c < ∞ such that \overline{F^{∗n}}(x) ≤ c(1 + ε)^{n}\overline{F} (x) for all n ≥ 2, x ≥ 0.

Remark The results stated in Lemmas 4.3 to 4.6 are well-known. For a proof of Lemmas 4.3 and 4.4, see the papers of Cline (1986) and Embrechts and Goldie (1982), for Lemma 4.5 see Pitman (1980). Lemma 4.6 is due to Kesten and can be found in Athreya and Ney (1972).

4.3 Asymptotics of Single Summands

We begin the proof of Theorem 4.1 with investigating the asymptotic behaviour of single summands appearing in (16).

Lemma 4.7 For x → ∞,

IP (Y_{1}^{0} + (r_{2} − 1)J_{1} > x, ξ_{1}^{0}(T_{1}) = 1, S_{2} > J_{1})
∼ IP(Y_{1}^{0} > x, ξ_{1}^{0}(T_{1}) = 1)IP(S > J) , (18)
IP ((r_{2} − 1)S_{n} + Y_{n}^{1} > x, ξ_{n}^{1}(T_{n}) = 0) ∼ IP(Y_{n}^{1} > x, ξ_{n}^{1}(T_{n}) = 0) , (19)
IP ((r_{2} − 1)S_{n} + Y_{n}^{1} + (r_{2} − 1)J_{n} > x, ξ_{n}^{1}(T_{n}) = 1, S_{n+1} > J_{n})
∼ IP(Y_{n}^{1} > x, ξ_{n}^{1}(T_{n}) = 1)IP(S > J) . (20)

Proof Note that

IP (Y_{1}^{0} + (r_{2} − 1)J_{1} > x, ξ_{1}^{0}(T_{1}) = 1, S_{2} > J_{1})
= IP (Y_{1}^{0} + (r_{2} − 1)J_{1} > x, ξ_{1}^{0}(T_{1}) = 1) IP(S > J) .

On the other hand, Lemmas 3.4 and 4.3 imply that the tail function \overline{F} (x) = IP (Y_{1}^{0} > x | ξ_{1}^{0}(T_{1}) = 1) belongs to S. Thus, having in mind that the tail of J_{1} is exponentially bounded, (18) follows from Lemma 4.5. The proof of (19) and (20) is analogous. □

4.4 Asymptotics of Finite Sums

We now investigate the asymptotics of finite sums of terms appearing in (16). For n ≥ 1, i = 0, 1, let A_{n}^{i} = {ξ_{n}^{i}(T_{n}) = 1, S_{n+1} < J_{n}}, B_{n}^{0} = {ξ_{n}^{1}(T_{n}) = 0} and B_{n}^{1} = {ξ_{n}^{1}(T_{n}) = 1, S_{n+1} > J_{n}}. For notational ease we sometimes drop the superscript i. We also write Y_{n} for Y_{n}^{1}.

Lemma 4.8 For x → ∞, i = 0, 1,

(a) IP(Y_{1}^{i} +Y_{2}^{1} > x, A_{1}^{i}, B_{2}^{0}) ∼ (π_{1}IP(S < J)IP(B_{2}^{0}) + IP(A_{1}^{i})π_{0}) IP(μT > x)

(b) IP(Y_{1}^{i} + Y_{2}^{1} > x, A_{1}^{i}, B_{2}^{1}) ∼ (π_{1}IP(S < J)IP(B_{2}^{1}) + IP(A_{1}^{i})π_{1}IP(S > J)) IP(μT > x).

Proof Note that IP(Y_{1}^{i} + Y_{2}^{1} > x, A_{1}^{i}, B_{2}^{0}) = IP(A_{1}^{i})IP(B_{2}^{0})\overline{G_{1} ∗ G_{2}}(x), where

\overline{G}_{1}(x) = \frac{IP(Y_{1}^{i} > x, A_{1}^{i})}{IP(A_{1}^{i})}, \overline{G}_{2}(x) = \frac{IP(Y_{2}^{1} > x, B_{2}^{0})}{IP(B_{2}^{0})} .

By the result of Lemma 3.4 we have

\overline{G}_{1}(x) ∼ \frac{π_{1}IP(S < J)}{IP(A_{1}^{i})}IP(μT > x), \overline{G}_{2}(x) ∼ \frac{π_{0}}{IP(B_{2}^{0})}IP(μT > x) .

Thus, since T is subexponential, Lemma 4.4 implies that

IP(Y_{1}^{i} + Y_{2}^{1} > x, A_{1}^{i}, B_{2}^{0}) ∼ (π_{1}IP(S < J)IP(B_{2}^{0}) + IP(A_{1}^{i})π_{0}) IP(μT > x)

and (a) is shown. The proof of (b) is analogous. □

Lemma 4.9 For n ≥ 2, i = 0, 1 and x → ∞,

(a) IP(Y_{1}^{i} + Y_{2} + … + Y_{n} > x, A_{1}^{i}, A_{2}^{1}, …, A_{n−1}^{1}, B_{n}^{0})

∼ (π_{1}IP(S < J)IP(A_{2}^{1})^{n−2}IP(B_{n}^{0})
+(n − 2)π_{1}IP(S < J)IP(A_{1}^{i})IP(A_{2}^{1})^{n−3}IP(B_{n}^{0})
+IP(A_{1}^{i})IP(A_{2}^{1})^{n−2}π_{0})IP(μT > x),

(b) IP(Y_{1}^{i} + Y_{2} + … + Y_{n} > x, A_{1}^{i}, A_{2}^{1}, …, A_{n−1}^{1}, B_{n}^{1})

∼ (π_{1}IP(S < J)IP(A_{2}^{1})^{n−2}IP(B_{n}^{1})
+(n − 2)π_{1}IP(S < J)IP(A_{1}^{i})IP(A_{2}^{1})^{n−3}IP(B_{n}^{1})

+IP(A_{1}^{i})IP(A_{2}^{1})^{n−2}π_{1}IP(S > J))IP(μT > x).

Proof We prove the lemma by induction with respect to n. For n = 2 this is just Lemma 4.8. Suppose that (a) holds for some n ≥ 2. Note that

IP(Y_{1}^{i} + Y_{2} + … + Y_{n+1} > x, A_{1}^{i}, A_{2}^{1}, …, A_{n}^{1}, B_{n+1}^{0})
= IP(A_{1}^{i})IP(A_{2}^{1})^{n−1}IP(B_{n+1}^{0})\overline{G_{1} ∗ G_{2}}(x) ,

where

\overline{G}_{1}(x) = \frac{IP(Y_{1}^{i} + Y_{2} + … + Y_{n−1} + Y_{n+1} > x, A_{1}^{i}, A_{2}^{1}, …, A_{n−1}^{1}, B_{n+1}^{0})}{IP(A_{1}^{i})IP(A_{2}^{1})^{n−2}IP(B_{n+1}^{0})}

and \overline{G}_{2}(x) = IP(Y_{n} > x, A_{n}^{1})(IP(A_{n}^{1}))^{−1}. Applying our hypothesis to \overline{G}_{1}(x) and the result of Lemma 3.4 to \overline{G}_{2}(x), Lemma 4.4 yields (a). The proof of (b) is analogous. □

4.5 Proof of Theorem 4.1

From representation formula (16) we get

IP(B > x) = IP ((r_{2} − 1)J_{0} > x, S_{0} > I_{0}, S_{1} > J_{0}) + IP(S < I)×

×[IP(Y_{1}^{0} > x, ξ_{1}^{0}(T_{1}) = 0) + IP(Y_{1}^{0} + (r_{2} − 1)J_{1} > x, ξ_{1}^{0}(T_{1}) = 1, S_{2} > J_{1})

+ ∑_{n=2}^{∞} {IP(Y_{1}^{0} + ∑_{j=2}^{n} ((r_{2} − 1)S_{j} + Y_{j}^{1}) > x, A_{1}^{0}, A_{2}^{1}, …, A_{n−1}^{1}, B_{n}^{0})

+IP(Y_{1}^{0} + ∑_{j=2}^{n} ((r_{2} − 1)S_{j} + Y_{j}^{1}) + (r_{2} − 1)J_{n} > x,

A_{1}^{0}, A_{2}^{1}, …, A_{n−1}^{1}, B_{n}^{1})}]

+ IP(S > I)IP(S < J)[IP ((r_{2} − 1)S_{1} + Y_{1}^{1} > x, ξ_{1}^{1}(T_{1}) = 0)

+IP ((r_{2} − 1)S_{1} + Y_{1}^{1} + (r_{2} − 1)J_{1} > x, ξ_{1}^{1}(T_{1}) = 1, S_{2} > J_{1})

+ ∑_{n=2}^{∞} {IP( ∑_{j=1}^{n} ((r_{2} − 1)S_{j} + Y_{j}^{1}) > x, A_{1}^{1}, …, A_{n−1}^{1}, B_{n}^{0})

+IP( ∑_{j=1}^{n} ((r_{2} − 1)S_{j} + Y_{j}^{1}) + (r_{2} − 1)J_{n} > x, A_{1}^{1}, …, A_{n−1}^{1}, B_{n}^{1})}]

Following the same argumentation as in the proof of Lemma 4.7 we can asymptotically neglect the S_{j} and J_{j} in all summands. Furthermore, using Lemma 4.6, we can pass with the limit x → ∞ under the infinite sum ∑_{n=2}^{∞}. With the notation p = IP(S < J), α_{i}^{j} = IP(ξ^{j}(T) = i) for i, j = 0, 1, by Lemma 4.9 we thus obtain the following result. Due to the fact that IP(B_{n}^{0}) + IP(B_{n}^{1}) = 1 − IP(A_{n}^{1}), for x → ∞,

IP(B > x) ∼ {IP(S < I)[(1 − π_{1}p) + ∑_{n=2}^{∞} (π_{1}p(α_{1}^{1}p)^{n−2}(1 − α_{1}^{1}p)

+(n − 2)π_{1}pα_{1}^{0}p(α_{1}^{1}p)^{n−3}(1 − α_{1}^{1}p) + α_{1}^{0}p(α_{1}^{1}p)^{n−2}(1 − π_{1}p))]

+IP(S > I)p[(1 − π_{1}p) + ∑_{n=2}^{∞} ((n − 1)π_{1}p(α_{1}^{1}p)^{n−2}(1 − α_{1}^{1}p)

+(α_{1}^{1}p)^{n−1}(1 − π_{1}p))]}IP(μT > x)

= (IP(S < I)\frac{1 − π_{1} − α_{1}^{1}p + 2π_{1}p − p}{(1 − π_{1})(1 − α_{1}^{1}p)} + \frac{p}{1 − α_{1}^{1}p})IP(μT > x) ,

where in the last equation we have used that π_{0}α_{1}^{0} + π_{1}α_{1}^{1} = π_{1}. Since IP(S < I) = ν/(ν + λ_{off}), π_{1} = λ_{off}/(λ_{on} + λ_{off}), p = ν/(ν + λ_{on}), we obtain

IP(S < I)\frac{1 − π_{1} − α_{1}^{1}p + 2π_{1}p − p}{(1 − π_{1})(1 − α_{1}^{1}p)} + \frac{p}{1 − α_{1}^{1}p}

= \frac{IP(S < I)}{1 − π_{1}} + (IP(S < I) \frac{2π_{1}p − π_{1} − p}{1 − π_{1}} + p) \frac{1}{1 − α_{1}^{1}p}

= \frac{IP(S < I)}{1 − π_{1}} = \frac{ν(λ_{on} + λ_{off})}{(ν + λ_{off})λ_{on}} .

Thus,

lim_{x→∞} \frac{IP(B > x)}{IP(μT > x)} = \frac{ν(λ_{on} + λ_{off})}{(ν + λ_{off})λ_{on}} . □

5 Fluid Model with m Sources

Suppose now that m ≥ 2. In this more general case we can use similar arguments as in Section 4 since a representation formula analogous to (16) can be derived for the generic net inflow B during an aggregate activity period. Let q ∈ {2, …, m}. The idea is to group together sources 1 to q to a single source with on-periods C^{(q)}, where C^{(q)} denotes the aggregate activity period of sources 1 to q, and off-periods Z^{(q)} = min(S, I^{(2)}, …, I^{(q)}), which are exponentially distributed with mean 1/(ν + ∑_{k=2}^{q} λ_{off}^{(k)}). Let B^{(q)} be the net inflow of all sources 1 to m during C^{(q)}. Then, with the notation

A_{q,n}^{i} = {ζ_{q,n}^{i}(C_{n}^{(q−1)}) = 1, Z_{n+1}^{(q−1)} < J_{n}^{(q)}} ,
B_{q,n}^{i,0} = {ζ_{q,n}^{i}(C_{n}^{(q−1)}) = 0} , B_{q,n}^{i,1} = {ζ_{q,n}^{i}(C_{n}^{(q−1)}) = 1, Z_{n+1}^{(q−1)} > J_{n}^{(q)}} ,

where {ζ_{q}^{i_{q}}(t)} is the (q − 1)-st component of {ξi(t)}, indicating whether at time t source q is active or not, we have

IP (B^{(q)} > x)
∼ IP (Z^{(q−1)} < I^{(q)}) [IP (B_{1}^{(q−1)} > x, B_{q,1}^{0,0}) + IP (B_{1}^{(q−1)} > x, B_{q,1}^{0,1})

+ ∑_{n=2}^{∞} {IP(B_{1}^{(q−1)} + … + B_{n}^{(q−1)} > x, A_{q,1}^{0}, A_{q,2}^{1}, …, A_{q,n−1}^{1}, B_{q,n}^{1,0})

+IP (B_{1}^{(q−1)} + … + B_{n}^{(q−1)} > x, A_{q,1}^{0}, A_{q,2}^{1}, …, A_{q,n−1}^{1}, B_{q,n}^{1,1}) }]

+ IP (Z^{(q−1)} > I^{(q)}) IP (Z^{(q−1)} < J^{(q)}) [IP (B^{(q−1)} > x, B_{q,1}^{1,0})
+IP (B^{(q−1)} > x, B_{q,1}^{1,1})  (21)

+ ∑_{n=2}^{∞} {IP(B_{1}^{(q−1)} + … + B_{n}^{(q−1)} > x, A_{q,1}^{1}, …, A_{q,n−1}^{1}, B_{q,n}^{1,0})

+IP(B_{1}^{(q−1)} + … + B_{n}^{(q−1)} > x, A_{q,1}^{1}, …, A_{q,n−1}^{1}, B_{q,n}^{1,1})}] .

From this we obtain the following result.

Theorem 5.1 For each q ∈ {2, …, m},

IP(B^{(q)} > x) ∼ \frac{ν}{ν + ∑_{k=2}^{q} λ_{off}^{(k)}} ∏_{k=2}^{q} (1 + \frac{λ_{off}^{(k)}}{λ_{on}^{(k)}} ) IP(μT > x) , x → ∞ . (22)

We prove Theorem 5.1 by induction with respect to the number q ∈ {2, …, m} of grouped sources. Note that our considerations in Section 4 do not depend on the fact that for m = 2 the process {η(t)} admits only the two states r_{1} − 1 and r_{1} − 1 + r_{2}. Thus, considering the process {η(t)} given in (7), we also obtain

IP(B^{(2)} > x) ∼ \frac{ν}{ν + λ_{off}^{(2)}} (1 + \frac{λ_{off}^{(2)}}{λ_{on}^{(2)}} ) IP(μT > x) , x → ∞ ,

where μ = lim_{t→∞} IE (r_{1} − 1 + ∑_{k=2}^{m} r_{k}1I(k ∈ ξ(t))). Now suppose that (22) holds for some q ∈ {2, …, m − 1}. Then, the distribution of B^{(q)} is subexponential and thus the lemmas stated in Section 4.2 can be applied to this distribution. Theorem 5.1 is proved if we show that (22) holds for q + 1 as well. For showing this we will make use of the representation formula (21) and therefore we need some auxiliary results on the asymptotical behaviour of the summands in (21).

5.1 Some Auxiliary Results

Let π̃_{q} = (π̃_{q,0}, π̃_{q,1}) be the stationary distribution of {ζ_{q}^{i}(t)}, i = 0, 1.

Lemma 5.1 For x → ∞, i, j = 0, 1,

IP (B^{(q−1)} > x, ζ_{q}^{i}(C^{(q−1)}) = j) ∼ π̃_{q,j}IP (B^{(q−1)} > x) .

Proof Similar as in the proof of Lemma 3.4 we have, with the abbreviating notation B = B^{(q−1)} and C = C^{(q−1)}, for some δ > 0 and for all x sufficiently large,

IP(B > x, ζ_{q}^{i}(C) = j) = IP(B > x, C > \frac{x}{r_{max}} , ζ_{q}^{i}(C) = j)

= ∫_{x}^{∞} ∫ ∞_{\frac{x}{rmax}} (p_{ij}(t) − π̃_{q,j} + π̃_{q,j}) dF_{(B,C)}(s, t)
≤ (e^{−δ \frac{x}{rmax}} + π̃_{q,j})IP(B > x, C > \frac{x}{r_{max}} )

and therefore

lim sup_{x→∞} \begin{matrix}IP & (B > x, ζ_{q}^{i}(C) = j) \\ IP & (B > x) \end{matrix} ≤ π̃_{q,j} .

On the other hand,

IP (B > x, ζ_{q}^{i}(C) = j) = ∫_{x}^{∞} ∫ ∞_{\frac{x}{rmax}} (p_{ij}(t) − π̃_{q,j} + π̃_{q,j}) dF_{(B,C)}(s, t)

≥ − ∫_{x}^{∞} ∫ ∞_{\frac{x}{rmax}} e^{−δt}dF_{(B,C)}(s, t) + π̃_{q,j}IP (B > x, C > \frac{x}{r_{max}} )
≥ −e^{−δ \frac{x}{rmax}} IP (B > x, C > \frac{x}{r_{max}} ) + π̃_{q,j}IP (B > x)

and thus

lim inf_{x→∞} \begin{matrix}IP & (B > x, ζ_{q}^{i}(C) = j) \\ IP & (B > x) \end{matrix} ≥ π̃_{q,j} . □

The following two lemmas are similar to Lemmas 4.8 and 4.9. Lemma 5.2 For x → ∞, i = 0, 1,

(a) IP (B_{1}^{(q−1)} + B_{2}^{(q−1)} > x, A_{q,1}^{i}, B_{q,2}^{1,0})

∼ (π̃_{q,1}IP(Z^{(q−1)} < J^{(q)})IP(B_{q,2}^{1,0}) + IP(A_{q,1}^{i})π̃_{q,0})IP(B^{(q−1)} > x),

(b) IP(B_{1}^{(q−1)} + B_{2}^{(q−1)} > x, A_{q,1}^{i}, B_{q,2}^{1,1}) ∼ (π̃_{q,1}IP(Z^{(q−1)} < J^{(q)})IP(B_{q,2}^{1,1}) +IP(A_{q,1}^{i})π̃_{q,1}IP(Z^{(q−1)} > J^{(q)}))IP(B^{(q−1)} > x).

Proof Note that

IP (B_{1}^{(q−1)} + B_{2}^{(q−1)} > x, A_{q,1}^{i}, B_{q,2}^{1,0}) = IP (A_{q,1}^{i}) IP (B_{q,2}^{1,0}) \overline{G_{1} ∗ G_{2}}(x) where

\overline{G}_{1}(x) = \begin{matrix}IP & (B_{1}^{(q−1)} > x, A_{q,1}^{i}) \\ IP & (A_{q,1}^{i}) \end{matrix} , \overline{G}_{2}(x) = \begin{matrix}IP & (B_{2}^{(q−1)} > x, B_{q,2}^{1,0}) \\ IP & (B_{q,2}^{1,0}) \end{matrix} .

By the result of Lemma 5.1 we have

\overline{G}_{1}(x) ∼ \begin{matrix}π̃_{q,1} & IP & (Z^{(q−1)} < J^{(q)}) \\ IP & (A_{q,1}^{i}) \end{matrix}IP (B^{(q−1)} > x)

and

\overline{G}_{2}(x) ∼ \frac{π̃_{q,0}}{IP (B_{q,2}^{1,0})}IP (B^{(q−1)} > x) .

Thus, Lemma 4.4 implies that

IP (B_{1}^{(q−1)} + B_{2}^{(q−1)} > x, A_{q,1}^{i}, B_{q,2}^{1,0})
∼ (π̃_{q,1}IP (Z^{(q−1)} < J^{(q)}) IP (B_{q,2}^{1,0}) + IP (A_{q,1}^{i}) π̃_{q,0}) IP (B^{(q−1)} > x)

and (a) is shown. The proof of (b) is analogous. □

Lemma 5.3 For n ≥ 2, i = 0, 1 and x → ∞,

(a) IP(B_{1}^{(q−1)} + … + B_{n}^{(q−1)} > x, A_{q,1}^{i}, A_{q,1}^{2}, …, A_{q,n−1}^{1}, B_{q,n}^{1,0}) ∼ (π̃_{q,1}IP(Z^{(q−1)} < J^{(q)})IP(A_{q,2}^{1})^{n−2}IP(B_{q,n}^{1,0})

+(n − 2)π̃_{q,1}IP(Z^{(q−1)} < J^{(q)})IP(A_{q,1}^{i})IP(A_{q,2}^{1})^{n−3}IP(B_{q,n}^{1,0})

+IP(A_{q,1}^{i})IP(A_{q,2}^{1})^{n−2}π̃_{q,0})IP(B^{(q−1)} > x),

(b) IP(B_{1}^{(q−1)} + B_{2}^{(q−1)} + … + B_{n}^{(q−1)} > x, A_{q,1}^{i}, A_{q,2}^{1}, …, A_{q,n−1}^{1}, B_{q,n}^{1,1}) ∼ (π̃_{q,1}IP(Z^{(q−1)} < J^{(q)})IP(A_{q,2}^{1})^{n−2}IP(B_{q,n}^{1,1})

+(n − 2)π̃_{q,1}IP(Z^{(q−1)} < J^{(q)})IP(A_{q,1}^{i})IP(A_{q,2}^{1})^{n−3}IP(B_{q,n}^{1,1}) +IP(A_{q,1}^{i})IP(A_{q,2}^{1})^{n−2}π̃_{q,1}IP(Z^{(q−1)} > J^{(q)}))IP(B^{(q−1)} > x).

The proof of this lemma is analogous to the proof of Lemma 4.9.

5.2 Proof of Theorem 5.1

We are now in the position to perform the induction step. We use the notation p̃_{q} = IP(Z^{(q−1)} < J^{(q)}) and α̃_{q,i}^{j} = IP(ζ_{q}^{j}(C^{(q−1)}) = i) for i, j = 0, 1. Then, due to the fact that IP(B_{q,n}^{1,0}) + IP(B_{q,n}^{1,1}) = 1 − IP(A_{q,n}^{1}), we obtain from representation formula (21) and Lemma 5.3, for x → ∞,

IP(B^{(q)} > x) ∼ (IP (Z^{(q−1)} < I^{(q)}) (1 + \frac{α̃_{q,1}^{0}p̃_{q}}{1 − α̃_{q,1}^{1}p̃_{q}} )

+IP (Z^{(q−1)} > I^{(q)}) p̃_{q} \frac{1}{1 − α̃_{q,1}^{1}p̃_{q}} )IP (B^{(q−1)} > x)

= (IP (Z^{(q−1)} < I^{(q)}) \frac{1 − π̃_{q,1} − α̃_{q,1}^{1}p̃_{q} + 2π̃_{q,1}p̃_{q} − p̃_{q}}{(1 − π̃_{q,1})(1 − α̃_{q,1}^{1}p̃_{q})} + \frac{p̃_{q}}{1 − α̃_{q,1}^{1}p̃_{q}} ) ×

×IP (B^{(q−1)} > x) , where in the last equation we have used that π̃_{q,0}α̃_{q,1}^{0} +π̃_{q,1}α̃_{q,1}^{1} = π̃_{q,1}. Since IP (Z^{(q−1)} < I^{(q)}) = (ν+∑_{k=2}^{q−1} λ_{off}^{(k)})/(ν+∑_{k=2}^{q} λ_{off}^{(k)}), π̃_{q,1} = λ_{off}^{(q)}/(λ_{on}^{(q)}+λ_{off}^{(q)})

and p̃_{q} = (ν + ∑_{k=2}^{q−1} λ_{off}^{(k)})/(ν + ∑_{k=2}^{q−1} λ_{off}^{(k)} + λ_{on}^{(q)}), we obtain

IP(Z^{(q−1)} < I^{(q)})\frac{1 − π̃_{q,1} − α̃_{q,1}^{1}p̃_{q} + 2π̃_{q,1}p̃_{q} − p̃_{q}}{(1 − π̃_{q,1})(1 − α̃_{q,1}^{1}p̃_{q})} + \frac{p̃_{q}}{1 − α̃_{q,1}^{1}p̃_{q}}

= \frac{IP(Z^{(q−1)} < I^{(q)})}{1 − π̃_{q,1}} + (IP(Z^{(q−1)} < I^{(q)}) \frac{2π̃_{q,1}p̃_{q} − π̃_{q,1} − p̃_{q}}{1 − π̃_{q,1}} + p̃_{q}) ×

×\frac{1}{1 − α̃_{q,1}^{1}p̃_{q}} = \frac{IP(Z^{(q−1)} < I^{(q)})}{1 − π̃_{q,1}} = \frac{ν + ∑_{k=2}^{q−1} λ_{off}^{(k)}}{ν + ∑_{k=2}^{q} λ_{off}^{(k)}} \frac{λ_{on}^{(q)} + λ_{off}^{(q)}}{λ_{on}^{(q)}} .

Thus, by the hypothesis of our induction, for x → ∞

IP(B^{(q)} > x) ∼ \frac{ν + ∑_{k=2}^{q−1} λ_{off}^{(k)}}{ν + ∑_{k=2}^{q} λ_{off}^{(k)}} \frac{λ_{on}^{(q)} + λ_{off}^{(q)}}{λ_{on}^{(q)}} IP (B^{(q−1)} > x)

∼ \frac{ν + ∑_{k=2}^{q−1} λ_{off}^{(k)}}{ν + ∑_{k=2}^{q} λ_{off}^{(k)}} \frac{λ_{on}^{(q)} + λ_{off}^{(q)}}{λ_{on}^{(q)}} \frac{ν}{ν + ∑_{k=2}^{q−1} λ_{off}^{(k)}} ∏_{k=2}^{q−1} (1 + \frac{λ_{off}^{(k)}}{λ_{on}^{(k)}} ) IP(μT > x)

= \frac{ν}{ν + ∑_{k=2}^{q} λ_{off}^{(k)}} ∏_{k=2}^{q} (1 + \frac{λ_{off}^{(k)}}{λ_{on}^{(k)}} ) IP(μT > x) ,

which means that (22) holds for q + 1. This completes the proof. □

6 Palm-Stationary Buffer Content Distribution

Recall that the development of the fluid flow queueing system can be seen as a random walk with generic increment B − min{S, I^{(2)}, …, I^{(m)}}. This enables us to determine the asymptotical behaviour of the tail of the steadystate distribution W (x) = lim_{n→∞} IP(W_{n} ≤ x) of the buffer content at the beginning of aggregate activity periods. However, in connection with this we have to determine the expected net input IEB during aggregate activity periods. From (22) we conclude that IEB < ∞ and hence IEC < ∞. The process {ξ(t)} with ξ(t) = ∑_{k=1}^{m} r_{k}1I(source k is active at time t) is regenerative and hence the stochastic mean value theorem (see Cohen (1976)) implies that the expected total input L into the system per unit time can be given in the following way:

L = \frac{IE ∫_{0}^{C+Z} ∑_{k=1}^{m} r_{k}1I(source k is active at time t) dt}{IE(C + Z)} ,

where L is given in (3) and Z = min{S, I^{(2)}, …, I^{(m)}}. Thus,

L = \frac{IEB + IEC}{IEC + IEZ} .  (23)

The same argument applied to the alternating renewal process with generic alternating variables Z and C yields

\frac{1}{1 + IET ν} ∏_{k=2}^{m} \frac{λ_{on}^{(k)}}{λ_{on}^{(k)} + λ_{off}^{(k)}} = \frac{IEZ}{IEC + IEZ} ,  (24)

where IEZ = 1/Λ = 1/(ν + ∑_{k=2}^{m} λ_{off}^{(k)}). Now, (23) and (24) imply that

IEB = \frac{1}{Λ} (1 − (1 − L)(1 + IET ν) ∏_{k=2}^{m} (1 + \frac{λ_{off}^{(k)}}{λ_{on}^{(k)}} )) ,  (25)

see also Cohen (1994). This, together with Theorem 5.1, allows us to determine the asymptotical behaviour of \overline{W} (x) = 1 − W (x) as x → ∞. By Theorem 5.1 we have

IP(B > x) ∼ \frac{ν}{Λ} ∏_{k=2}^{m} (1 + \frac{λ_{off}^{(k)}}{λ_{on}^{(k)}} ) IP(μT > x) , x → ∞ ,  (26)

i.e. the distribution F_{B} of the net input B during aggregate activity periods belongs to IR since F ∈ IR. Furthermore, we need the following closure property of class IR. Let F be the distribution of a nonnegative random variable with finite positive expectation μ. The distribution F^{s} given by

F^{s}(x) = \frac{1}{μ} ∫_{0}^{x} \overline{F} (y) dy,  x ≥ 0  (27)

is called the integrated tail distribution of F.

Lemma 6.1 If F ∈ IR, then F^{s} ∈ IR.

The proof easily follows from the defining equations (1) and (27).

Theorem 6.1 Let L denote the quantity given in (3). If F ∈ IR, then

\overline{W} (x) ∼ \frac{ν}{(1 − L)(1 + IET ν)} ∫_{x}^{∞} IP(μT > u)du , x → ∞ .  (28)

Proof From Lemmas 4.2 and 6.1 we conclude that F_{B}^{s} is subexponential. Then, a well-known result on the asymptotical tail behaviour of the maximum of random walks with negative drift (see Embrechts and Veraverbeke (1982), Pakes (1975)) gives, together with (26),

\overline{W} (x) ∼ \frac{ν + ∑_{k=2}^{m} λ_{off}^{(k)}}{1 − (ν + ∑_{k=2}^{m} λ_{off}^{(k)}) IEB} ∫_{x}^{∞} IP(B > u)du

∼ \frac{Λ}{1 − ΛIEB} \frac{ν}{Λ} ∏_{k=2}^{m} (1 + \frac{λ_{off}^{(k)}}{λ_{on}^{(k)}} ) ∫_{x}^{∞} IP(μT > u)du .

Now, (28) follows from (25). □

Corollary 6.1 If \overline{F} is regular varying at infinity, i.e. \overline{F} (x) ∼ (α − 1)(IET)^{α} x^{−α}l(x), x → ∞, α > 1 and l(x) slowly varying at infinity, then

\overline{W} (x) ∼ \frac{ν}{(1 − L)(1 + νIET)}M^{α}x^{−(α−1)}l(x) , x → ∞ ,  (29)

where L is given in (3) and M = IET (r_{1} − 1 + ∑_{k=2}^{m}(r_{k}λ_{off}^{(k)})/(λ_{on}^{(k)} + λ_{off}^{(k)})).

Proof By Karamata’s theorem (see Bingham, Goldie and Teugels (1987))

∫_{x}^{∞} IP(μT > u)du = μ\frac{∫_{x/μ}^{∞} IP(T > u)du}{(x/μ)IP(T > (x/μ))} \frac{x}{μ}IP(T > (x/μ))

∼ 1/(α − 1)x(α − 1)(IET)^{α}(x/μ)^{−α}l(x/μ) ∼ (IET μ)^{α}x^{−(α−1)}l(x) ,

and thus (29) follows from (28). □

Remark Note that Corollary 6.1 is a special case of a result which has been recently obtained in Boxma (1997). As mentioned before, the result in Boxma (1997) concerns more general activity period distributions of sources 2, …, m, namely such with an exponentially bounded tail.

References

[1] Anantharam, V. (1995) On the sojourn time of sessions at an ATM buffer with longe-range dependent input traffic. Proc. 34th IEEE Conference on Decision and Control, December 1995.

[2] Athreya, K.B. and P.E. Ney (1972) Branching Processes. Springer– Verlag, Berlin.

[3] Bingham, N.H., C.M. Goldie and J.L. Teugels (1987) Regular Variation. Cambridge University Press. Cambridge.

[4] Boxma, O.J. (1996) Fluid queues and regular variation. Performance Evaluation 27/28, 699–712.

[5] Boxma, O.J. (1997) Regular variation in a multi-source fluid queue. In V. Ramaswami and P.E. Wirth, editors, Teletraffic Contributions for the Information Age (Proc. ITC 15), 391–402. Elsevier, Amsterdam.

[6] Choudhury, G.L. and W. Whitt (1997) Long-tail buffer-content distributions in broadband networks. Performance Evaluation 30, 177–190.

[7] Cline, D.B.H. (1986) Convolution tails, product tails and domains of attraction. Probability Theory and Related Fields 72, 529–557.

[8] Cline, D.B.H. (1994) Intermediate regular and Π variation. Proc. London Math. Soc. 68, 594–616.

[9] Cohen, J.W. (1976) On Regenerative Processes in Queueing Theory. Lect. Notes in Econ. Math. Syst. 121, Springer–Verlag, Berlin.

[10] Cohen, J.W. (1994) On the effective bandwidth in buffer design for the multi-server channels. Report BS-R9406. CWI Amsterdam.

[11] Embrechts, P. and C.M. Goldie (1982) On convolution tails. Stochastic Processes and Their Applications 13, 263–278.

[12] Embrechts, P., C.M. Goldie and N. Veraverbeke (1979) Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitstheorie verw. Geb. 49, 335–347.

[13] Embrechts, P. and N. Veraverbeke (1982) Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72.

[14] Heath, D., S. Resnick and G. Samorodnitsky (1996a) Heavy tails and long range dependence in on/off processes and associated fluid models. Mathematics of Operations Research (to appear).

[15] Heath, D., S. Resnick and G. Samorodnitsky (1996b) Patterns of buffer overflow in a class of queues with long memory in the input stream. Ann. Appl. Prob. (to appear).

[16] Heyman, D.P. and T.V. Lakshman (1996) Source models for VBR broadcast-video traffic. IEEE J. Select. Areas in Communications 4, 40–48.

[17] Iosifescu, M. (1980) Finite Markov Chains and Their Applications. J. Wiley & Sons, New York.

[18] Jelenković, P.R. and Lazar, A.A. (1996) Asymptotic results for multiplexing subexponential on-off sources. Techn Report 457-96-23. Columbia University, New York.

[19] Pakes, A.G. (1975) On the tails of waiting-time distribution. J. Appl. Probability 12, 555–564.

[20] Palmowski, Z. and T. Rolski (1996) A note on martingale inequalities for fluid models. Statistics & Probability Letters 31, 13–21.

[21] Pitman, E.J.G. (1980) Subexponential distribution functions. J. Austral. Math. Soc. (Series A) 29, 337-347.

[22] Whitt, W. (1993) Tail probabilities with statistical multiplexing and effective bandwidths in multi-class queues. Telecommunication Systems 2, 71–107.