Loss rates for stochastic fluid models

Abstract

We introduce loss rates, a novel class of performance measures for Markovian stochastic fluid models and discuss their applications potential. We derive analytical expressions for loss rates and describe efficient methods for their evaluation. Further, we study interesting asymptotic properties of loss rates for large size of the buffer, which are crucial for identifying the Quality of Service requirements guaranteed for each user. We illustrate the theory with a numerical example.

Introduction

Let $\{\phi \left(t\right),t\ge 0\}$ be an irreducible continuous-time Markov Chain (CTMC) with a finite state space $\mathcal{S}=\{1,2,\dots ,n\}$ and infinitesimal generator $T$. We assume that the CTMC is positive recurrent with stationary distribution vector $\pi$. Let $\{(\phi \left(t\right),X\left(t\right)),t\ge 0\}$ be a Markovian stochastic fluid model (SFM) with phase variable $\phi \left(t\right)$, level variable $X\left(t\right)$, a lower boundary $X\left(t\right)\ge 0$, an upper boundary $X\left(t\right)\le B$, and real rates $c_i$ for all $i\in \mathcal{S}$, such that

• when $0<X\left(t\right)<B$ and $\phi \left(t\right)=i$, then the rate at which the level is changing is $c_i$,

• when $X\left(t\right)=0$ and $\phi \left(t\right)=i$, then the rate at which the level is changing is $max\{c_i,0\}$, and

• when $X\left(t\right)=B$ and $\phi \left(t\right)=i$, then the rate at which the level is changing is $min\{c_i,0\}$.

That is, when the SFM hits level zero, which must occur in some phase $j$ with $c_j<0$, it will stay on level zero until a transition to some phase $k$ with $c_k>0$ occurs. Similarly, when the SFM hits level $B$, which must occur in some phase $j$ with $c_j>0$, it will stay on level $B$ until a transition to some phase $k$ with $c_k<0$ occurs. The buffer collecting fluid in this process is denoted by $X$. The CTMC is referred to as the driving process.

We partition the set of all phases as $\mathcal{S}={\mathcal{S}}_1\cup {\mathcal{S}}_2\cup {\mathcal{S}}_0$, where ${\mathcal{S}}_1=\{i\in \mathcal{S}:c_i>0\},{\mathcal{S}}_2=\{i\in \mathcal{S}:c_i<0\},{\mathcal{S}}_0=\{i\in \mathcal{S}:c_i=0\}$, and generator as

$$T=\left[\begin{array}{ccc}\hfill T_{11}\hfill & \hfill T_{12}\hfill & \hfill T_{10}\hfill \\ \hfill T_{21}\hfill & \hfill T_{22}\hfill & \hfill T_{20}\hfill \\ \hfill T_{01}\hfill & \hfill T_{02}\hfill & \hfill T_{00}\hfill \end{array}\right]\text{,}$$

according to the partitioning of $\mathcal{S}$.

The analytical expressions for the stationary and transient analysis of the SFMs have been derived in the literature, and powerful algorithms exist for the numerical evaluations of various performance measures  [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. The class of bounded SFMs that we study here, has been earlier analyzed in series of papers of Latouche and Da Silva Soares, who analyzed the steady-state distribution of the buffer content, see  [11], [12], [13] for example. Also, alternative approaches for the steady-state solution have been developed in  [14], [15]. The transient analysis of the model has been derived by Bean, O’Reilly and Taylor  [16].

In this paper, we focus on the transient analysis of the SFMs and extend it to the study of other interesting performance measures of this model. Specifically, we introduce loss rates, a novel class of performance measures for stochastic fluid models and discuss their applications potential. The notion of loss rate was already analyzed in many contexts: in Lévy driven queues  [17], [18], in a single server queue  [19], [20], [21], risk theory  [22] etc. Also, other performance measures were introduced in  [23].

First, we introduce relative-time loss rates, a measure of the time spent at the boundary $B$ with respect to the duration of the busy period, as follows. Let $\theta \left(x\right)=inf\{t>0:X\left(t\right)=x\}$ be the first passage time to level $x$. Define matrix $E^{\theta \left(0\right)}={\left[E_{i,j}^{\theta \left(0\right)}\right]}_{i\in {\mathcal{S}}_1,j\in {\mathcal{S}}_2}$, such that the $(i,j)$ entry is given by

$$E_{i,j}^{\theta \left(0\right)}=E\left[\theta \left(0\right)I(\phi \left(\theta \left(0\right)\right)=j)\right|\phi \left(0\right)=i,X\left(0\right)=0]\text{,}$$

and interpreted as the mean first passage time to level zero and doing so in phase $j$, given start in level zero and phase $i$. Let $\tau \left(t\right)={\int }_0^{t}I(X\left(u\right)=B)du$ be the total time spent on the upper boundary $B$ up to time $t$. Define matrix $E^{\tau \left(\theta \left(0\right)\right)}={\left[E_{i,j}^{\tau \left(\theta \left(0\right)\right)}\right]}_{i\in {\mathcal{S}}_1,j\in {\mathcal{S}}_2}$ such that the $(i,j)$ entry is given by

$$E_{i,j}^{\tau \left(\theta \left(0\right)\right)}=E\left[\tau \left(\theta \left(0\right)\right)I(\phi \left(\theta \left(0\right)\right)=j)\right|\phi \left(0\right)=i,X\left(0\right)=0]\text{,}$$

and interpreted as the mean time spent at the boundary $B$ before visiting level zero and doing so in phase $j$, given start in level zero and phase  $i$. Now, for all $i\in {\mathcal{S}}_1,j\in {\mathcal{S}}_2$ we define relative-time loss rate $Z_{ij}$ by

$$Z_{i,j}=\frac{E_{i,j}^{\tau \left(\theta \left(0\right)\right)}}{E_{i,j}^{\theta \left(0\right)}}\text{.}$$

We note that the information about the time spent at the boundary $B$ may not be sufficient for the evaluation of the system, since no fluid is lost during the times when the process is at the boundary $B$ in some phase $i$ with $c_i=0$. In particular, it is possible for the process to be spending significant times at the boundary $B$ without necessarily losing a lot of fluid, and so we are interested in a measure that captures the information about the lost fluid. Such loss occurs whenever the buffer $X$ is full and in some phase $k$ with positive rate $c_k>0$. Consequently, we introduce the absolute-volume loss rates, which is a measure of total fluid lost during the busy period. Let $V\left(t\right)={\int }_0^t{c}_{\phi \left(u\right)}I(X\left(u\right)=B)I(c_{\phi \left(u\right)}>0)du$ be the total fluid volume lost up to time $t$. Define matrix $E^{V\left(\theta \left(0\right)\right)}={\left[E_{i,j}^{V\left(\theta \left(0\right)\right)}\right]}_{i\in {\mathcal{S}}_1,j\in {\mathcal{S}}_2}$, such that the $(i,j)$ entry is given by

$$E_{i,j}^{V\left(\theta \left(0\right)\right)}=E\left[V\left(\theta \left(0\right)\right)I(\phi \left(\theta \left(0\right)\right)=j)\right|\phi \left(0\right)=i,X\left(0\right)=0]\text{,}$$

and interpreted as the mean total fluid volume lost at the moment of the first return to level zero and doing so in phase $j$, given start in level zero and phase $i$. Now, for all $i\in {\mathcal{S}}_1,j\in {\mathcal{S}}_2$ we define absolute-volume loss rate $M_{ij}$ by

$$M_{i,j}=\frac{E_{i,j}^{V\left(\theta \left(0\right)\right)}}{E_{i,j}^{\theta \left(0\right)}}\text{.}$$

We note that, a large amount of lost fluid and a large amount of fluid entering the buffer may be observed simultaneously. Therefore, it would be useful to have a measure that captures the information about the proportion of the fluid lost. Consequently, we introduce relative-volume loss rates which are the measure of the total fluid lost with respect to the total fluid that entered the buffer during the busy period. Let $W\left(t\right)={\int }_0^t{c}_{\phi \left(u\right)}I(c_{\phi \left(u\right)}>0)du$ be the total fluid volume that has flowed into buffer $X$ up to time $t$. Define matrix $E^{W\left(\theta \left(0\right)\right)}={\left[E_{i,j}^{W\left(\theta \left(0\right)\right)}\right]}_{i\in {\mathcal{S}}_1,j\in {\mathcal{S}}_2}$, such that the $(i,j)$ entry is given by

$$E_{i,j}^{W\left(\theta \left(0\right)\right)}=E\left[W\left(\theta \left(0\right)\right)I(\phi \left(\theta \left(0\right)\right)=j)\right|\phi \left(0\right)=i,X\left(0\right)=0]\text{,}$$

and interpreted as the mean total fluid that has flowed into buffer $X$ up to the moment of the first return to level zero and doing so in phase $j$, given start in level zero and phase $i$.

Remark 1

Here, we apply the terminology used within the theory of SFMs, so that $c_i>0$ corresponds to the fluid entering, and $c_i<0$ to fluid leaving the buffer. Naturally, when considering the class of related models referred to as fluid queues, in which $a_i$ is the rate of fluid entering the buffer, $b_i$ is the rate of fluid leaving the buffer, and $c_i=a_i-b_i$ is the net input rate, then within such setting we can interpret $W\left(t\right)$ as the mean total net fluid that has flowed into buffer $X$.

Now, for all $i\in {\mathcal{S}}_1,j\in {\mathcal{S}}_2$ we define relative-volume loss rate $M_{ij}^{\ast }$ by

$$M_{i,j}^{\ast }=\frac{E_{i,j}^{V\left(\theta \left(0\right)\right)}}{E_{i,j}^{W\left(\theta \left(0\right)\right)}}\text{.}$$

We derive analytical expressions for above loss rates and describe efficient methods for their evaluation. Further, we study interesting asymptotic properties of loss rates and derive explicit expressions which can be conveniently evaluated even for systems of large size. This is crucial for identifying the Quality of Service (QoS) requirements guaranteed for each user. If there are not enough network resources, that is when buffer content exceeds buffer size $B$, the connection is rejected. The quantities considered in this paper give information on the proportion of the lost information which could be set on certain level by choosing the buffer size $B$ large enough.

The paper is organized as follows. In Section  2 we summarize standard results for SFMs. In Section  3 we derive expressions for loss rates, and in Section  4 we analyze them for large buffer  $B$. In Section  5 we illustrate our results in the context of a numerical example. Throughout we assume that $\Re \left(s\right)>0$ in all definitions of the Laplace–Stieltjes transforms (LSTs).

We emphasize that the loss rates introduced in this paper are transient measures, which form a part of the transient analysis of the SFMs. An approach for the loss rates in the stationary sense have been developed in  [24]. Both transient and stationary analyses play an important role in the theory of SFMs. Here, we illustrate that the loss rates in the transient sense, can also be used to gain useful insights about the dynamics of the process, in particular during the stage when the stationary behavior may have not been reached yet. Specifically, we show that different initial settings (starting phase) of the system may have an impact on the loss rates, and hence on the performance of the system.