Performance bounds for feedforward queueing networks with upper-constrained inputs

Abstract

We propose a simple framework for analyzing feedforward queueing networks that have the following features: each customer belongs to a (customer) flow and the route that a customer follows in the network depends on the flow to which he belongs. The amount of traffic fed by a customer flow into the network is bounded by a deterministic subadditive function, which is called a subadditive envelope in this paper. Customers are served according to a FIFO discipline at each queue. We show that the boundary of the virtual-waiting-time distribution at each queue in such networks can be analytically derived, based on the information concerning subadditive envelopes of flows. By using the proposed analysis, we investigate how the envelope of a flow changes when it traverses a series of queues.

Introduction

Queueing networks are often used as mathematical tools for modeling computer networks, where queues represent buffers in switching nodes or routers and arcs between queues are communication links. Although queuing networks are generally difficult to analyze, there is a set of queueing networks that are analytically tractable. For example, Baskett, Chandy, Muntz, and Palacios found a set of multiclass queueing networks, known as BCMP networks, which have product-form equilibrium distributions under four types of service disciplines: last-in-first-out (LIFO), processor sharing (PS), infinite servers (IS), and first-in-first-out (FIFO) [1], [2]. Kelly found that general multiclass networks, known as Kelly networks, which connect a set of quasi-reversible queues under very general routing schemes, have product-form equilibrium distributions [3], [4], [5]. In these networks, customers arrive from outside according to a Poisson process. Each arriving customer is independently routed in a network. Each customer requires different service times at separate queues on his route and these service times are statistically independent from each other. A customer may visit the same queue several times and the network might be closed. The route of a customer in the network is stochastically determined on a hop-by-hop basis.

Most current telecommunication networks, including the Internet, are packet-switched networks, which might initially seem to be well suited to queueing network models. Current telecommunication networks are, however, quite different from BCMP or Kelly networks. For example, it has been reported that packet-arrival processes in the Internet are far from Poisson processes. The size of a packet is usually fixed during the transmission in the network, and thus the service times received by a given packet at separate queues are strongly correlated. A packet visits each queue at most once and the path followed by a packet in the network is uniquely determined according to its destination. Thus, it is natural to construct types of queueing networks that have features typical to the current telecommunication networks and to seek a mathematical framework for analyzing such networks.

In this paper, we propose a method for analyzing networks that have the following features:

• Feedforward network [2], [5]: A customer after his service at a queue goes only to a downstream queue.

• Flow of customers: Each customer belongs to a specific flow. The route followed by a customer in the network depends on the flow to which he belongs.

• Upper constrained input: The upper limit for the amount of traffic fed by a flow is governed by a deterministic subadditive function, which we refer to as a subadditive envelope. The arrival process or service-time distribution (job-size distribution) of customers is arbitrary besides this constraint.

• FIFO-service discipline with superadditive-service curve: Customers are served according to a first-in-first-out (FIFO) discipline at each queue. The lower limit for the amount of traffic served at a queue during a given interval is governed by a deterministic superadditive function.

These features correspond well to those of the Internet. For example, a flow of customers corresponds to an application flow established between end hosts. The amount of traffic fed by a host (flow) has a deterministic bound because end hosts are connected to the Internet via finite-bandwidth access lines. If the traffic from an end host is regulated by a leaky-bucket-based regulator, then the amount of traffic fed by the host is upper bounded by a subadditive deterministic function. The route of a packet in the Internet depends on the destination IP address (and thus the flow to which the packet belongs).

An important difference between traditional queueing networks and the above mentioned networks is in feature (4) (upper constrained input). In the traditional queueing networks, some stochastic models are used to describe the customer arrival processes and job size distributions. Meanwhile, feature (4) implies that the traffic is described only through the subadditive envelope. Since the subadditive envelope is the maximum amount of traffic fed by a flow within a given period, actual packet arrival processes or job-size distributions are not uniquely determined. Thus, most existing works concerning queueing networks with upper-constrained inputs are focused on deriving a limit to the waiting time at each queue.

In this paper, we show that, for networks having features (1)–(4), a bound of the complementary cumulative distribution function (CCDF) of the virtual-waiting time at each queue is analytically tractable. In particular, we show how to numerically evaluate a bound of the CCDF of the virtual-waiting time; based on the information concerning the subadditive envelopes of flows. The queueing networks we consider in this paper include the IntServ [6] and DiffServ [7] architectures, which have been proposed for quality-of-service management in the Internet, and thus we can analyze the queueing delay characteristics of IntServ- or DiffServ-based networks based on our proposal.

The analysis of queueing networks with upper-constrained inputs have been addressed by several authors [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. Cruz [11], [12] derived the worst-case delay bound for acyclic networks when the amount of traffic from each flow is leaky-bucket constrained. Parekh and Gallager [15], [16] extended Cruz’s results to cases where some scheduling algorithms are used at queues. Their results hold also for cyclic network topologies. Stiliadis [17] and Agrawal et al. [8] have derived the worst-case delay bound for general upper-constrained flows. They also introduced the service-curve concept for modeling various scheduling algorithms used in network nodes. Chang [9] and Toyoizumi [18] consider the worst-case delay bound when flows share buffers and bandwidth. These studies did not consider the delay distribution. Several works [10], [13], [14], [19] focused on a bound of virtual-waiting-time (or backlog) distribution at a single-stage queue where various upper-constrained flows are multiplexed. Most works, however, did not consider queueing networks. Vojnović and Boudec [20] studied a stochastic analysis of a queueing network with upper-constrained inputs, but they assumed that the subadditive envelope of a flow remains unchanged when traversing queues.

The remainder of the paper is organized as follows. In Section 2, we set up our model of the network and specify notation. In Section 3, we derive the maximum waiting time and a bound of the CCDF of the virtual-waiting time for a single-stage queue, in which various upper-constrained flows are multiplexed together. We also show how the output process of each flow is upper-constrained, and derive the expression for the subadditive envelope of each individual output flow. In Section 4, we extend the results in Section 3 to cover general feedforward networks. In Section 5, we numerically show the tightness of the bound of the virtual-waiting-time CCDF. By using the proposed analysis, we also investigate how the envelope of a flow changes when it traverses a series of queues. Finally, Section 6 concludes with some remarks.