Multi-Hop Probing Asymptotics in Available Bandwidth Estimation:
Stochastic Analysis

Xiliang Liu1, Kaliappa Ravindran2, Dmitri Loguinov3

1 City University of New York, New York, NY 10016  

2 City College of New York, New York, NY 10031  

3 Texas A&M University, College Station, TX 77843

 

Abstract

This paper analyzes the asymptotic behavior of packet-train probing over a multi-hop network path $ \mathcal{P}$carrying arbitrarily routed bursty cross-traffic flows. We examine the statistical mean of the packet-train output dispersions and its relationship to the input dispersion. We call this relationship the response curve of path $ \mathcal{P}$. We show that the real response curve $ \mathcal{Z}$is tightly lower-bounded by its multi-hop fluid counterpart $ \mathcal{F}$, obtained when every cross-traffic flow on $ \mathcal{P}$is hypothetically replaced with a constant-rate fluid flow of the same average intensity and routing pattern. The real curve $ \mathcal{Z}$asymptotically approaches its fluid counterpart $ \mathcal{F}$as probing packet size or packet train length increases. Most existing measurement techniques are based upon the single-hop fluid curve $ \mathcal{S}$associated with the bottleneck link in $ \mathcal{P}$. We note that the curve $ \mathcal{S}$coincides with $ \mathcal{F}$in a certain large-dispersion input range, but falls below $ \mathcal{F}$in the remaining small-dispersion input ranges. As an implication of these findings, we show that bursty cross-traffic in multi-hop paths causes negative bias (asymptotic underestimation) to most existing techniques. This bias can be mitigated by reducing the deviation of $ \mathcal{Z}$from $ \mathcal{S}$using large packet size or long packet-trains. However, the bias is not completely removable for the techniques that use the portion of $ \mathcal{S}$that falls below $ \mathcal{F}$.

1 Introduction

End-to-end estimation of the spare capacity along a network path using packet-train probing has recently become an important Internet measurement research area. Several measurement techniques such as TOPP [14], Pathload [6], IGI/PTR [5], Pathchirp [16], and Spruce [17] have been developed. Most of the current proposals use a single-hop path with constant-rate fluid cross-traffic to justify their methods. The behavior and performance of these techniques in a multi-hop path with general bursty cross-traffic is limited to experimental evaluations. Recent work [9] initiated the effort of developing an analytical foundation for bandwidth measurement techniques. Such a foundation is important in that it helps achieve a clear understanding of both the validity and the inadequacy of current techniques and provides a guideline to improve them. However, the analysis in [9] is restricted to single-hop paths. There is still a void to fill in understanding packet-train bandwidth estimation over a multi-hop network path.

Recall that the available bandwidth of a network hop is its residual capacity after transmitting cross-traffic within a certain time interval. This metric varies over time as well as a wide range of observation time intervals. However, in this paper, we explicitly target the measurement of a long-term average available bandwidth, which is a stable metric independent of observation time instances and observation time intervals [9]. Consider an $ N$-hop network path $ \mathcal{P}=(L_1,L_2,\ldots,L_N)$, where the capacity of link $ L_i$is denoted by $ C_i$and the long-term average of the cross-traffic arrival rate at $ L_i$is given by $ \lambda_i$, which is assumed to be less than $ C_i$. The hop available bandwidth of $ L_i$is $ A_i=C_i-\lambda_i$. The path available bandwidth $ A_{\mathcal{P}}$is given by

$\displaystyle A_{\mathcal{P}}=\min_{1\le i \le N}(C_i-\lambda_i).$

(1)


The hop $ L_b$, which carries the minimum available bandwidth, is called the tight link or the bottleneck linkIn general, the tight link can be different from the link with the minimum capacity, which we refer to as the narrow link of $ \mathcal{P}$.. That is,

$\displaystyle b=\mathrm{arg} \min_{1\le i\le N}(C_i-\lambda_i).$

(2)


The main idea of packet-train bandwidth estimation is to infer $ A_{\mathcal{P}}$from the relationship between the inter-packet dispersions of the output packet-trains and those of the input packet-trains. Due to the complexity of this relationship in arbitrary network paths with bursty cross-traffic flows, previous work simplifies the analysis using a single-hop path with fluidWe use the term ``fluid" and ``constant-rate fluid" interchangeably. cross-traffic, while making the following two assumptions without formal justification: first, cross-traffic burstiness only causes measurement variability that can be smoothed out by averaging multiple probing samples and second, non-bottleneck links have negligible impact on the proposed techniques.

The validity of the first assumption is partially addressed in [9], where the authors use a single-hop path with bursty cross-traffic to derive the statistical mean of the packet-train output dispersions as a function of the input probing dispersion, referred to as the single-hop response curve. Their analysis shows that besides measurement variability, cross-traffic burstiness can also cause measurement bias to the techniques that are based on fluid analysis. This measurement bias cannot be reduced even when an infinite number of probing samples are used, but can be mitigated using long packet-trains and/or large probing packet size.

This paper addresses further the two assumptions that current techniques are based on. To this end, we extend the asymptotic analysis in [9] to arbitrary network paths and uncover the nature of the measurement bias caused by bursty cross-traffic flows in a multi-hop network path. This problem is significantly different from previous single-hop analysis due to the following reasons. First, unlike single-hop measurements, where the input packet-trains have deterministic and equal inter-packet separation formed by the probing source, the input packet-trains at any hop (except the first one) along a multi-link path are output from the previous hop and have random structure. Second and more importantly, the multi-hop probing asymptotics are strongly related to the routing pattern of cross-traffic flows. This issue never arises in a single-hop path and it has received little attention in prior investigation. However, as we show in this paper, it is one of the most significant factors that affect the accuracy of bandwidth measurement in multi-hop paths.

To characterize packet-train bandwidth estimation in its most general settings, we derive the probing response curve $ \mathcal{Z}$of a multi-hop path $ \mathcal{P}$assuming arbitrarily routed bursty cross-traffic flows. We compare $ \mathcal{Z}$with its multi-hop fluid counterpart $ \mathcal{F}$, which is a response curve obtained when every cross-traffic flow in $ \mathcal{P}$is hypothetically replaced with a fluid flow of the same average intensity and routing pattern. We show, under an ergodic stationarity assumption for each cross-traffic flow, that the real curve $ \mathcal{Z}$is tightly lower bounded by its fluid counterpart $ \mathcal{F}$and that the curve $ \mathcal{Z}$asymptotically approaches its fluid bound $ \mathcal{F}$in the entire input range as probing packet size or packet-train length increases.

Most of the existing techniques are based on the single-hop fluid response curve $ \mathcal{S}$associated with the bottleneck link in $ \mathcal{P}$. Therefore, any deviation of the real curve $ \mathcal{Z}$from the single-hop curve $ \mathcal{S}$can potentially cause measurement bias in bandwidth estimation. Note that the deviation $ \mathcal{Z}-\mathcal{S}$can be decomposed as

$\displaystyle \mathcal{Z}-\mathcal{S}=(\mathcal{Z}-\mathcal{F})+(\mathcal{F}-\mathcal{S}).$

(3)


The first term $ \mathcal{Z}-\mathcal{F}$is always positive and causes asymptotic underestimation of $ A_\mathcal{P}$for most of the existing techniques. This deviation term and its resulting measurement bias are ``elastic" in the sense that they can be reduced to a negligible level using packet-trains of sufficient lengthThe analysis assumes infinite buffer space at each router.. For the second deviation term $ \mathcal{F}-\mathcal{S}$, we note that both $ \mathcal{S}$and $ \mathcal{F}$are piece-wise linear curves. The first two linear segments in $ \mathcal{F}$associated with large input dispersions coincide with $ \mathcal{S}$(i.e., $ \mathcal{F}-\mathcal{S}=0$). The rest of the linear segments in $ \mathcal{F}$associated with small input dispersions appear above $ \mathcal{S}$(i.e., $ \mathcal{F}-\mathcal{S}>0$). The amount of deviation and the additional negative measurement bias it causes are dependent on the routing patterns of cross-traffic flows, and are maximized when every flow traverses only one hop along the path (which is often called one-hop persistent cross-traffic routing [4]). Furthermore, the curve deviation $ \mathcal{F}-\mathcal{S}$is ``non-elastic" and stays constant with respect to probing packet size and packet-train length at any given input rate. Therefore, the measurement bias it causes cannot be overcome by adjusting the input packet-train parameters.

Among current measurement techniques, pathload and PTR operate in the input probing range where $ \mathcal{F}$coincides with $ \mathcal{S}$, and consequently are only subject to the measurement bias caused by the first deviation term $ \mathcal{Z}-\mathcal{F}$. Spruce may use the probing range where $ \mathcal{F}-\mathcal{S}>0$. Hence it is subject to both elastic and non-elastic negative measurement biases. The amount of bias can be substantially more than the actual available bandwidth in certain common scenarios, leading to negative results by the measurement algorithm and a final estimate of zero by the tool.

The rest of the paper is organized as follows. Section 2 derives the multi-hop response curve $ \mathcal{F}$assuming arbitrarily routed fluid cross-traffic flows and examines the deviation term $ \mathcal{F}-\mathcal{S}$. In Section 3 and 4, we derive the real response curve $ \mathcal{Z}$of a multi-hop path and show its relationship to its fluid counterpart $ \mathcal{F}$. We provide practical evidence for our theoretical results using testbed experiments and real Internet measurements in Section 5. We examine the impact of these results on existing techniques in Section 6 and summarize related work in Section 7. Finally, we briefly discuss future work and conclude in Section 8.

Due to limited space, most of the proofs in this paper are omitted, and we refer interested readers to [10] for more technical details.


2 Multi-Hop Fluid Analysis

It is important to first thoroughly understand the response curve $ \mathcal{F}$of a network path carrying fluid cross-traffic flows, since as we show later, the fluid curve $ \mathcal{F}$is an approachable bound of the real response curve $ \mathcal{Z}$. Initial investigation of the fluid curves is due to Melandar et al. [13] and Dovrolis et al. [3]. However, prior work only considers two special cross-traffic routing cases (one-hop persistent routing and path persistent routing). In this section, we formulate and solve the problem for arbitrary cross-traffic routing patterns, based on which, we discuss several important properties of the fluid response curves that allow us to obtain the path available bandwidth information.

2.1 Formulating A Multi-Hop Path

We first introduce necessary notations to formulate a multi-hop path and the cross-traffic flows that traverse along the path.

An $ N$-hop network path $ \mathcal{P}=(L_1,L_2,\ldots,L_N)$is a sequence of $ N$interconnected First-Come First-Served (FCFS) store-and-forward hops. For each forwarding hop $ L_i$in $ \mathcal{P}$, we denote its link capacity by $ C_i$, and assume that it has infinite buffer space and a work-conserving queuing discipline. Suppose that there are $ M$fluid cross-traffic flows traversing path $ \mathcal{P}$. The rate of flow $ j$is denoted by $ x_j$and the flow rate vector is given by $ \mathbf{x}=(x_1,x_2,\ldots,x_M)$.

We impose two routing constraints on cross-traffic flows to simplify the discussion. The first constraint requires every flow to have a different routing pattern. In the case of otherwise, the flows with the same routing pattern should be aggregated into one single flow. The second routing constraint requires every flow to have only one link where it enters the path and also have only one (downstream) link where it exits from the path. In the case of otherwise, the flow is decomposed into several separate flows that meet this routing constraint.

Definition 1   A flow aggregation is a set of flows, represented by a ``selection vector" $ \mathbf{p}=(p_1, p_2, \ldots , p_M)^T$, where $ p_j=1$if flow $ j$belongs to the aggregation and $ p_j=0$if otherwise. We use $ \mathbf{f}_j$to represent the selection vector of the aggregation that contains flow $ j$alone.

There are several operations between flow aggregations. First, the common flows to aggregations $ \mathbf{p}$and $ \mathbf{q}$form another aggregation, whose selection vector is given by $ \mathbf{p}\odot\mathbf{q}$, where the operator $ \odot$represents ``element-wise multiplication." Second, the aggregation that contains the flows in $ \mathbf{p}$but not in $ \mathbf{q}$is given by $ \mathbf{p}-\mathbf{p}\odot\mathbf{q}$. Finally, note that the traffic intensity of aggregation $ \mathbf{p}$can be computed from the inner product $ \mathbf{x}\mathbf{p}$.

We now define several types of flow aggregation frequently used in this paper. First, the traversing flow aggregation at link $ L_i$, denoted by its selection vector $ \mathbf{r}_i$, includes all fluid flows that pass through $ L_i$. The $ M\times N$matrix $ \mathbf{R}=(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N)$becomes the routing matrix of path $ \mathcal{P}$. For convenience, we define an auxiliary selection vector $ \mathbf{r}_0=\mathbf{0}$.

The second type of flow aggregation, denoted by $ \mathbf{e}_i$, includes all flows entering the path at link $ L_i$, which can be expressed as $ \mathbf{e}_i=\mathbf{r}_i-\mathbf{r}_i\odot\mathbf{r}_{i-1}$given the second routing constraint stated previously. The third type of flow aggregation, which includes flows that enter the path at link $ L_k$and traverse the downstream link $ L_i$, is denoted as $ \Gamma_{k,i}=\mathbf{e}_k\odot\mathbf{r}_i$, where $ k\le i$.

The cross-traffic intensity at link $ L_i$is denoted by $ \lambda_i$. We assume $ \lambda_i<C_i$for $ 1\le i\le N$. Since none of the links in $ \mathcal{P}$is congested, the arrival rate of flow $ j$at any link it traverses is $ x_j$. Consequently, we have

$\displaystyle \lambda_i=\mathbf{x}\mathbf{r}_{i}<C_i, \quad 1\le i\le N.$

(4)


We further define the path configuration of $ \mathcal{P}$as the following $ 2\times N$matrix

$\displaystyle \mathbf{H}=\left(\begin{array}{cccc} C_1&C_2&\ldots&C_N\ \lambda_1&\lambda_2&\ldots&\lambda_N \end{array}\right ).$

(5)


The hop available bandwidth of $ L_i$is given by $ A_i=C_i-\lambda_i$. We assume that every hop has different available bandwidth, and consequently that the tight link is unique. Sometimes, we also need to refer to the second minimum hop available bandwidth and the associated link, which we denote as $ A_{b2}=C_{b2}-\lambda_{b2}$and $ L_{b2}$, respectively. That is

$\displaystyle b2=\mathrm{arg}\min_{1\le i \le N, i\neq b}(C_i-\lambda_i),$

(6)


where $ b$is the index of the tight hop.

2.2 Fluid Response Curves

We now consider a packet-train of input dispersion (i.e., inter-packet spacing) $ g_I$and packet size $ s$that is used to probe path $ \mathcal{P}$. We are interested in computing the output dispersion of the packet train and examining its relation to $ g_I$. Such a relation is called the gap response curve of path $ \mathcal{P}$. It is easy to verify that under fluid conditions, the response curve does not depend on the packet-train length $ n$. Hence, we only consider the case of packet-pair probing. We denote the output dispersion at link $ L_i$as $ \gamma_i(g_I,s)$or $ \gamma_i$for short, and again for notational convenience we let $ \gamma_0=g_I$. Note that $ \gamma_N(g_I,s)$corresponds to the notation $ \mathcal{F}$we have used previously.

Based on our formulations, the gap response curve of path $ \mathcal{P}$has a recursive representation given below.

Theorem 1   When a packet-pair with input dispersion $ g_I$and packet size $ s$is used to probe an $ N$-hop fluid path with routing matrix $ \mathbf{R}$and flow rate vector $ \mathbf{x}$, the output dispersion at link $ L_i$can be recursively expressed as

$\displaystyle \gamma_i=\begin{cases}g_I & i=0\ \max\left(\gamma_{i-1},\dfrac{s+\Omega_i} {C_i}\right)&i>0 \end{cases},$

(7)


where $ \Omega_i$is The term $ \Omega_i$represents the volume of fluid cross-traffic buffered between the packet-pair in the outgoing queue of link $ L_i$. For an analogical understanding, we can view the packet-pair as a bus, the cross-traffic as passengers, and the routers as bus stations. Then, $ \Omega_i$is the amount of cross-traffic picked up by the packet-pair at link $ L_i$as well as all the upstream links of $ L_i$. This cross-traffic will traverse over link $ L_i$due to the flows' routing decision.

$\displaystyle \Omega_i=\sum_{k=1}^{i}\Bigl[\gamma_{k-1}\mathbf{x}\Gamma_{k,i} \Bigr].$

(8)


Proof. Assumes that the first probing packet arrives at link $ L_i$at time instance $ a_1$. It gets immediate transmission service and departs at $ a_1+s/C_i$. The second packet arrives at $ a_1+\gamma_{i-1}$. The server of $ L_i$needs to transmit $ s+\Omega_i$amount of data before it can serve the second packet. If this is done before time instance $ a_1+\gamma_{i-1}$, the second packet also gets immediate service and $ \gamma_i=\gamma_{i-1}$. Otherwise, the sever undergoes a busy period between the departure of the two packets, meaning that $ \gamma_i=(s+\Omega_i)/C_i$. Therefore, we have

$\displaystyle \gamma_i=\max\left(\gamma_{i-1},\dfrac{s+\Omega_i}{C_i}\right).$

(9)


This completes the proof of the theorem. $ \qedsymbol$

As a quick sanity check, we verify the compatibility between Theorem 1 and the special one-hop persistent routing case, where every flow that enters the path at link $ L_i$will exit the path at link $ L_{i+1}$. For this routing pattern, we have

$\displaystyle \Gamma_{k,i}=\begin{cases}\mathbf{0} &i\neq k \ \mathbf{r}_i &i=k \end{cases}.$

(10)


Therefore, equation (8) can be simplified as

$\displaystyle \Omega_i=\gamma_{i-1}\mathbf{x}\mathbf{r}_i=\gamma_{i-1}\lambda_i,$

(11)


which agrees with previous results [3], [13].

2.3 Properties of Fluid Response Curves

Theorem 1 leads to several important properties of the fluid response curve $ \mathcal{F}$, which we discuss next. These properties tell us how bandwidth information can be extracted from the curve $ \mathcal{F}$, and also show the deviation of $ \mathcal{F}$, as one should be aware of, from the single-hop fluid curve $ \mathcal{S}$of the tight link.

Property 1   The output dispersion $ \gamma_N(g_I,s)$is a continuous piece-wise linear function of the input dispersion $ g_I$in the input dispersion range $ (0,\infty)$.

Let $ 0=\alpha_{K+1}<\alpha_K<\ldots<\alpha_1<\alpha_0=\infty$be the input dispersion turning points that split the gap response curve to $ K+1$linear segmentsNote that the turning points in $ \mathcal{F}$is indexed according to the decreasing order of their values. The reason will be clear shortly when we discuss the rate response curve.. Our next result discusses the turning points and linear segments that are of major importance in bandwidth estimation.

Property 2   The first turning point $ \alpha_1$corresponds to the path available bandwidth in the sense that $ A_{\mathcal{P}}=s/\alpha_1$. The first linear segment in the input dispersion range $ (\alpha_1=s/A_{\mathcal{P}},\infty)$has slope 1 and intercept 0. The second linear segment in the input dispersion range $ (\alpha_2, \alpha_1)$has slope $ \lambda_b/C_b$and intercept $ s/C_b$, where $ b$is the index of the tight link:

$\displaystyle \gamma_N(g_I,s)=\begin{cases} g_I & \alpha_1 \le g_I \le \infty \ \dfrac{g_I\lambda_b+s}{C_b} & \alpha_2 \le g_I \le \alpha_1 \end{cases}.$

(12)


These facts are irrespective of the routing matrix.

It helps to find the expression for the turning point $ \alpha_2$, so that we can identify the exact range for the second linear segment. However, unlike $ \alpha_1$, the turning point $ \alpha_2$is dependent on the routing matrix. In fact, all other turning points are dependent on the routing matrix and can not be computed based on the path configuration matrix alone. Therefore, we only provide a bound for $ \alpha_2$.

Property 3   For any routing matrix, the term $ s/\alpha_2$is no less than $ A_{b2}$, which is the second minimum hop available bandwidth of path $ \mathcal{P}$.

The slopes and intercepts for all but the first two linear segments are related to the routing matrix. We skip the derivation of their expressions, but instead provide both a lower bound and an upper bound for the entire response curve.

Property 4   For a given path configuration matrix, the gap response curve associated with any routing matrix is lower bounded by the single-hop gap response curve of the tight link

$\displaystyle \mathcal{S}(g_I,s)= \begin{cases} g_I & g_I>\dfrac{s}{A_{\mathca... ... \dfrac{s+g_I\lambda_b}{C_b} & 0< g_I <\dfrac{s}{A_{\mathcal{P}}} \end{cases}.$

(13)


It is upper bounded by the gap response curve associated with one-hop persistent routing.

We now make several observations regarding the deviation of $ \gamma_N(g_I,s)$(i.e., $ \mathcal{F}$) from $ \mathcal{S}(g_I,s)$. Combing (12) and (13), we see that $ \gamma_N(g_I,s)-\mathcal{S}(g_I,s)=0$when $ g_I\ge \alpha_2$. That is, the first two linear segments on $ \mathcal{F}$coincide with $ \mathcal{S}$. When $ g_I<\alpha_2$, Property 4 implies that the deviation $ \gamma_N(g_I,s)-\mathcal{S}(g_I,s)$is positive. The exact value depends on cross-traffic routing and it is maximized in one-hop persistent routing for any given path configuration matrix.

Also note that there are three pieces of path information that we can extract from the gap response curve $ \mathcal{F}$without knowing the routing matrix. By locating the first turning point $ \alpha_1$, we can compute the path available bandwidth. From the second linear segment, we can obtain the tight link capacity and cross-traffic intensity (and consequently, the bottleneck link utilization) information. Other parts of the response curve $ \mathcal{F}$are less readily usable due to their dependence on cross-traffic routing.

2.4 Rate Response Curves

To extract bandwidth information from the output dispersion $ \gamma_N$, it is often more helpful to look at the rate response curve, i.e., the functional relation between the output rate $ r_O=s/\gamma_N$and the input rate $ r_I=s/g_I$. However, since this relation is not linear, we adopt a transformed version first proposed by Melander et al. [14], which depicts the relation between the ratio $ r_I/r_O$and $ r_I$. Denoting this rate response curve by $ \tilde{\mathcal{F}}(r_I)$, we have

$\displaystyle \tilde{\mathcal{F}}(r_I)=\dfrac{r_I}{r_O}=\dfrac{\gamma_N(g_I,s)}{g_I}.$

(14)


This transformed version of the rate response curve is also piece-wise linear. It is easy to see that the first turning point in the rate curve is $ s/\alpha_1=A_p$and that the rate curve in the input rate range $ (0,s/\alpha_2)$can be expressed as

$\displaystyle \tilde{\mathcal{F}}(r_I)=\begin{cases} 1 & r_I\le A_{\mathcal{P}... ...ambda_b+r_I}{C_b} & \dfrac{s}{\alpha_2}\ge r_I\ge A_{\mathcal{P}} \end{cases}.$

(15)


Finally, it is also important to notice that the rate response curve $ \tilde{\mathcal{F}}(r_I)$does not depend on the probing packet size $ s$. This is because, for any given input rate $ r_I$, both $ \gamma_N(g_I,s)$and $ g_I$are proportional to $ s$. Consequently, the ratio between these two terms remains a constant for any $ s$.

2.5 Examples

We use a simple example to illustrate the properties of the fluid response curves. Suppose that we have a 3-hop path with equal capacity $ C_i=10$mb/s, $ i=1,2,3$. We consider two routing matrices and flow rate settings that lead to the same link load at each hop.

In the first setting, the flow rate vector $ \mathbf{x}=(4,7,8)$and the routing pattern is one-hop persistent, i.e., $ \mathbf{R}=$diag$ (1,1,1)$. In the second setting, the flow rate vector $ \mathbf{x}=(4,3,1)$and the routing pattern is path persistent. That is,

$\displaystyle \mathbf{R}=\left(\begin{array}{ccc} 1&1&1\ 0&1&1\ 0&0&1 \end{array}\right ).$

(16)


Both of the settings result in the same path configuration matrix

$\displaystyle \mathbf{H}=\left(\begin{array}{ccc} 10&10&10\ 4&7&8 \end{array}\right ).$

(17)


The probing packet size $ s$is $ 1500$bytes. The fluid gap response curves for the two routing patterns are plotted in Fig. 1(a). In this example, both curves have 4 linear segments separated by turning points $ \alpha_1=6$ms, $ \alpha_2=4$ms, and $ \alpha_3=2$ms. Note that part of the curve for path-persistent routing appears below the one for one-hop persistent routing. The lower bound $ \mathcal{S}$identified in Property 4 is also plotted in the figure. This lower bound is the gap response curve of the single-hop path comprising only the tight link $ L_3$.

\includegraphics[width=3.0in]{crfg.eps}             \includegraphics[width=3.0in]{crfr.eps} 

    (a) Gap response curves                                                                (b) Rate response curves

 

Figure 1: An example of multi-hop response curves.

The rate response curves for the two examples are given in Fig. 1(b), where the three turning points are $ 2$mb/s, $ 3$mb/s, and $ 6$mb/s respectively. Due to the transformation we adopted, the rate curve for one-hop persistent routing still remains as an upper bound for the rate curves associated with the other routing patterns. From Fig. 1(b), we also see that, similar to the gap curves, the two multi-hop rate response curves and their lower bound $ \tilde{\mathcal{S}}(r_I)$(i.e., the transformed rate version of $ \mathcal{S}(g_I,s)$) share the same first and second linear segments.

2.6 Discussion

We conclude this section by discussing several major challenges in extending the response curve analysis to a multi-hop path carrying bursty cross-traffic flows. First, notice that with bursty cross-traffic, even when the input dispersion and packet-train parameters remain constant, the output dispersion becomes random, rather than deterministic as in fluid cross-traffic. The gap response curve $ \mathcal{Z}$, defined as the functional relation between the statistical mean of the output dispersion and the input dispersion, is much more difficult to penetrate than the fluid curve $ \mathcal{F}$. Second, unlike in the fluid case, where both packet-train length $ n$and probing packet size $ s$have no impact on the rate response curve $ \tilde{\mathcal{F}}(r_I)$, the response curves in bursty cross-traffic are strongly related to these two packet-train parameters. Finally, a full characterization of a fluid flow only requires one parameter - its arrival rate, while a full characterization of a bursty flow requires several stochastic processes. In what follows, we address these problems and extend our analysis to multi-hop paths with bursty cross-traffic.


3 Basics of Non-Fluid Analysis

In this section, we present a stochastic formulation of the multi-hop bandwidth measurement problem and derive a recursive expression for the output dispersion random variable. This expression is a fundamental result that the asymptotic analysis in Section 4 is based upon.

3.1 Formulating Bursty Flows

We keep most of the notations the same as in the previous section, although some of the terms are extended to have a different meaning, which we explain shortly. Since cross-traffic flows now become bursty flows of data packets, we adopt the definitions of several random processes (Definition 1-6) in [9] to characterize them. However, these definitions need to be refined to be specific to a given router and flow aggregation. In what follows, we only give the definitions of two random processes and skip the others. The notations for all six random processes are given in Table 3.1.

Definition 2   The cumulative traffic arrival process of flow aggregation $ \mathbf{p}$at link $ L_i$, denoted as $ \{ V_i(\mathbf{p}, t ), 0 \le t < \infty\}$is a random process counting the total amount of data (in bits) received by hop $ L_i$from flow aggregation $ \mathbf{p}$up to time instance $ t$.

Definition 3   Hop workload process of $ L_i$with respect to flow aggregation $ \mathbf{p}$, denoted as $ \{W_i(\mathbf{p},t), 0 \le t < \infty\}$indicates the sum at time instance $ t$of service times of all packets in the queue and the remaining service time of the packet in service, assuming that flow aggregation $ \mathbf{p}$is the only traffic passing through link $ L_i$.

We next make several modeling assumptions on cross-traffic flows. First, we assume that all flows have stationary arrivals.

Assumption 1   For any cross-traffic flow $ j$that enters the path from link $ L_i$, the cumulative traffic arrival process $ \{V_i(\mathbf{f}_j,t)\}$has ergodic stationary increments. That is, for any $ \delta>0$, the $ \delta$-interval traffic intensity process $ \{Y_{i,\delta}(\mathbf{f}_j,t)\}$is a mean-square ergodic process with time-invariant distribution and ensemble mean $ x_j$.

We explain this assumption in more details. First, the stationary increment assumption implies that the increment process of $ \{V_i(\mathbf{f}_j,t)\}$for any given time interval $ \delta$, namely $ \{V_i(\mathbf{f}_j,t+\delta)-V_i(\mathbf{f}_j,t)=\delta Y_{i,\delta}(\mathbf{f}_j,t)\}$, has a time-invariant distribution. This further implies that the $ \delta$-interval traffic intensity process $ \{Y_{i,\delta}(\mathbf{f}_j,t)\}$is identically distributed, whose marginal distribution at any time instance $ t$can be described by the same random variable $ Y_{i,\delta}(\mathbf{f}_j)$. Second, the mean-square ergodicity implies that, as the observation interval $ \delta$increases, the random variable $ Y_{i,\delta}(\mathbf{f}_j)$converges to $ x_j$in the mean-square sense. In other words, the variance of $ Y_{i,\delta}(\mathbf{f}_j)$decays to 0 as