Bounding Delays in Packet-routing Networks with Light Traac Bounding Delays in Packet-routing Networks with Light T Raac (original) (raw)

Bounding delays in packet-routing networks

Proceedings of the twenty-seventh annual ACM symposium on Theory of computing - STOC '95, 1995

We consider the problem of computing the average packet delay in a general dynamic packet-routing network with Poisson input stream, during steady-state.

In Network of Queues, M/M/1 Can Outperform M/D/1

Let N be an open queueing network where the servers have generally distributed service times (with possibly different means) and the outside arrivals to the servers are Poisson. Define NC,FCFS (respectively, NE,FCFS ) to be queueing network N where each server has a constant (respectively, exponentially distributed) service time with the same mean as the corresponding server in N , and the packets are served in a First-Come-First-Served order. It has long been conjectured that for all networks N , the average packet delay in NC,FCFS is upper bounded by the average packet delay in NE,FCFS . In this paper, we present a counterexample to this conjecture

A Note on Stochastic Bounds for Queueing Networks

Advances in Applied Probability, 1984

Recently, Massey [1] proved that the vector of queue lengths of some queueing networks is stochastically dominated at any given time by that of a corresponding system of parallel M/M/l queues. This result is interesting, even though the bounds are generally quite conservative, in that the transient behavior of independent parallel M/M/l queues is considerably easier to analyze than that of a network.This note provides an alternative proof of a generalized form of that result.

Two Classes of Performance Bounds for Closed Queueing Networks

Performance Evaluation, 1987

Two classes of performance bounds for separable queueing networks are presented, one for single-chain networks and one for multichain networks. Unlike most bounds for single-chain networks, our bounds are not based upon the consideration of balanced networks. Instead, they are obtained by assuming mean queue lengths to be proportional to server loads; hence, they are called proportional bounds. Proportional bounds are tighter than balanced bounds because individual server loads are retained as parameters in a bound's formula. For the same reason, they require more computational effort than balanced bounds. We also show how proportional bounds are related to balanced bounds. Next we present generalized bounds that are calculated iteratively over sequences of population sizes; our method extends that of . These generalized bounds are shown to have a nested property. Furthermore, we present optimal population sequences, over all sequences of the same length, for getting the tightest upper and lower bounds. The other emphasis of this paper is on performance bounds for networks with many closed chains and many service centers. Bounding techniques are especially important for multichain networks since the computation time and space requirements are often so large that an exact solution is not feasible. Models of communication networks typically have many routing chains which are characterized by a sparseness property. In the computation of our performance bounds for multichain networks, we improve their accuracy by making use of routing information and exploiting the sparseness property. Proof o| Corollary 3.3. We only have to prove that rj(ni_l) Fro(hi_l) =qj(n,-1) qm(n,-1) J J for all i, 1 ~< i < S and all j, 1 ~<j ~< M. The proof is similar to that of Theorem 3.2, [] Proof of Theorem 3.4. We only have to prove that the upper bound obtained from population sequence Lemma A.4. For any i, 1 ~ i <~ S, and all j, 1 ~ j <~ M if ยง.(n,-1) rm(ni_l)<..qj(ni-1) qm(n~-1), J J then M M E qm(ni-11 1) E rm(n _O, m =j m =j [ Mmy'~=jr,,qm(ni--1)]

Constant Beats Memoryless for Service Times in a Markovian Queueing Network

We prove that a Markovian Open Queueing Network in which the service times are constant has lower average packet delay than the same network in which the service times are exponential (with the same mean). The proof is elementary, generalizing a similar result of Stamoulis and Tsitsiklis by removing the requirement that the network be layered. 1 Motivation Many real-world packet-routing network algorithms in which packets are routed to random destinations can be modeled by Markovian 1 Jackson Queueing Networks (M.J.Q.N.), except for the fact that the real-world networks require the service times at the servers to all be equal and constant [HBB94], [ST91]. In this paper we show that the average delay for a M.J.Q.N. with constant service times is upper-bounded by the average delay for the corresponding traditional M.J.Q.N. (with exponential service times). [ST91] proved this result for layered networks. Our proof exactly parallels the [ST91] proof, except that whereas their proof used...

An approximation to the response time for shortest queue routing

1989

In this paper we derive an approximation for the mean response time of a multiple queue system in which shortest queue routing is used. We assume there are K identical queues with infinite capacity and service times that are exponentially distributed. Arrivals of jobs to this system are Poisson and are routed to a queue of minimal length. We develop an approximation which is based on both theoretical and experimental considerations and, for K 5 8, has an relative error of less than one half of one percent when compared to simulation. For K = 16, the relative error is still acceptable, being less than 2 percent. An application to a model of parallel processing and a comparison of static and dynamic load balancing schemes are presented.

An Approximate Analytical Method for General Queueing Networks

IEEE Transactions on Software Engineering, 2000

In this paper, we present an approximate solution for the asymptotic behavior of relatively general queueing networks. In the particular case of networks with general service time distributions (i.e., fixed routing matrix, one or many servers per station, FIFO discipline), the application of the method gives relatively accurate results in a very short time. The approximate stationary state probabilities are identified with the solution of a nonlinear system. The proposed method is applicable to a larger class of queueing networks (dependent routing matrix, stations with fimite capacity, etc.). In this case, the structure of the network studied must satisfy certain decomposability conditions.

Routing in queues with delayed information

2003

We compare two routing-control strategies in a high-speed communication network with c parallel channels (routes), where information on service completions in down-stream servers is randomly delayed. The controller can either hold arriving messages in a common buffer, dispatching them to servers only when the delayed information becomes available (Wait option), or route jobs to the various channels, in a round-robin fashion, immediately upon their arrival. Interpreting the delays as servers's vacations and considering overall queue sizes as a measure of performance, we show that the Wait strategy is superior as long as the mean information delay is below a threshold. We calculate threshold values for various combinations of load and c and show that, for a given load, the threshold increases with c and, for fixed c, the threshold decreases with an increasing load. If information is delayed on arrival instants, rather than on service completions, we show that the system can be viewed as a tandem queue and derive a generalization of a queue-decomposition result obtained by Altman, Kofman and Yechiali.

Simple bounds for closed queueing networks

Queueing Systems - Theory and Applications - QUESTA, 1999

Consider a closed Jackson type network in which each queue has a single exponential server. Assume that N customers are moving among k queues. We establish simple closed form bounds on the network throughput (both lower and upper), which are sharper than those that are currently available. Numerical evaluation indicates that the improvements are significant.

On the nonconcavity of throughput in certain closed queueing networks

Performance Evaluation, 1986

Analytic queueing network models are being used to analyze various optimization problems such as server allocation, design and capacity issues, optimal routing, and workload allocation. The mathematical properties of the relevant performance measures, such as throughput, are important for optimization purposes and for insight into system performance.