Tandem Queueing Network Research Articles

Symmetry property of the throughput in closed tandem queueing networks with finite buffers

In this paper we consider closed tandem queueing networks with finite buffers and blocking before service. With this type of blocking, a server is allowed to start processing a job only if there is an empty space in the next buffer. It was recently conjectured that the throughput of such networks is symmetrical with respect to the population of the network. That is, the throughput of the network with population N is the same as that with population C − N, where C is the total number of buffer spaces in the network. The main purpose of this paper is to prove this result in the case where the service time distributions are of phase type (PH-distribution). The proof is based on the comparison of the sample paths of the network with populations N and C − N. Finally, we also show that this symmetry property is related to a reversibility property of this class of networks.

An Analytic Congestion Model for Closed Production Systems with IFR Processing Times

We present an analytic model relating the mean cycle time or throughput as a function of the number of jobs in a closed production system composed of a tandem network of queues having exponential and/or IFR processing times. This model exhibits macroscopic behavior that is predicted by results from queueing theory and involves three meaningful parameters: the bottleneck rate and the “raw process” time that can be determined from first moment data; and a dimensionless congestion coefficient that is typically obtained from a single WIP/average cycle time observation (e.g., simulation). The derivation of the model is based on observations of the behavior of the relation between mean cycle time and WIP. This “engineering” approach is different from a purely “probabilistic” one in that we do not consider the distribution of processing times at individual station. We compare the accuracy of the model in predicting mean cycle times to other techniques such as mean value analysis and simulation.

Strong consistency of infinitesimal perturbation analysis for tandem queueing networks

Strong consistency of infinitesimal perturbation analysis for the sojourn times in a class of tandem queueing networks is proved. Service times at the queues are correlated, and they are affine functions of the variable parameters. Differentiability of the average sojourn times is not assumed, but proved. The analysis is not based on assumptions of regenerative cycles of the networks but on stability and ergodicity of the queueing processes involved. The proof of strong consistency is based on a set of abstract conditions, described in terms of properties of the sample performance functions. These conditions are first shown to be sufficient for strong consistency, and then their validity for the networks in question is proved.

Comparisons of service disciplines in a tandem queueing network with real time constraints

In this paper we study the extremal properties of the stationary customer lag times in tandem G/GI/1 networks under different service disciplines in terms of convex and increasing convex orderings. Each customer carries a reference time with it and the lag time is defined to be the difference between the time that the customer departs from the system and its reference time. We show that among the class of work conserving non-preemptive service disciplines that are service time independent, the service discipline that schedules customers with the smallest reference times (SR) and the service discipline that schedules the customer with the largest reference time (LR) provide the minima and maxima respectively. If we restrict ourselves to the subset of these disciplines that do not use reference times but do use arrival times in making scheduling decisions, then the first in first out (FIFO) and last come first serve (LCFS) service disciplines provide the minima and maxima respectively. We also present similar results for G/M/1 queue when preemptions are allowed and for the class of service disciplines that are not work conserving.

Evaluation of the effects of protocol processing overhead in error recovery schemes for a high-speed packet switched network: link-by-link versus edge-to-edge schemes

The authors investigate the effects of protocol processing overhead on the performance of error recovery schemes. The focus is on the edge-to-edge error recovery scheme, in which retransmissions of erred packets only take place between source and destination nodes. An approximation is obtained for the Laplace transform for the distribution of the end-to-end packet transfer delay, considering the processing time required for error recovery. The performance of the link-by-link error recovery scheme, in which retransmissions take place between adjacent nodes, is evaluated and compared to the performance of the edge-to-edge scheme. Numerical results from a tandem queuing network model show that for a network with very-high-speed/low-error-rate channels, an edge-to-edge scheme gives a smaller packet transmission delay than a link-by-link scheme for both go-back-N and selective-repeat retransmission procedures, while keeping the packet loss probability sufficiently small.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An approximate analysis of open tandem queueing networks with blocking and general service times

An approximation algorithm is presented for open tandem queueing networks with finite buffers and with general service times. The algorithm decomposes the system into individual queues with revised arrival and service processes and revised queue capacity. Then, each queue is analyzed in isolation. The service process is revised to reflect the additional delay a unit might undergo due to blocking. Unlike previous algorithms, the arrival process to each decomposed queue is described by a C 2 distribution. The parameters of the service and the arrival processes are computed approximately using an iterative scheme. The approximation procedure yields the steady-state queue-length distribution of each queue. From this, other more commonly sought performance measures, such as mean queue-length, probability that a queue is empty, throughput, etc., can be easily computed. Comparisons of the approximate results with simulation results showed that the proposed algorithm has a good error-level.

On the asymptotic execution time of multi-tasked processes on tandem processors

Consider an ordered set of processes, each consisting of K ϵ Z + tasks, to be processed by a system of K processors in tandem. Each process, immediately after havings its i- th (1≤i<K) task served by the i- th processor, is queued up on a first-come-first-served basis for processing of its (i+1)- st task by the (i+1)- st processor, until all its tasks have been served. Define the execution time to be the time to serve all the processes in the set. Given that the processes form a stationary and ergodic sequence, as far as the processing times of the tasks are concernec, the asymptotic execution time is first explicitly computed, as the number of processes tends to infinity. This is done by using subadditive ergodic theory and certain results from the stability theory of tandem queueing networks. Then, design issues of such systems are discussed.

Optimal and Approximately Optimal Control Policies for Queues in Heavy Traffic

The “approximately” optimal control problem for tandem queueing or production networks (with local feedback allowed) under heavy traffic is treated. The buffers (scaled with traffic) are finite. The controls allow various inputs, connecting links, and the processors to be shut down or opened to manage the system. The service and arrival rates, as well as the routing probabilities, can also be controlled, and the system statistics can depend on the system state (scaled buffer occupancies). The associated costs involve holding costs, costs for shutting off/turning on the links or processors and the opportunity cost for lost production. It is shown that the (scaled) controlled system converges weakly (in an appropriate sense) to a controlled limit “reflected” diffusion. In the resealed time, the actions of the controllers lead to multiple “simultaneous” impulses in the limit problem. Thus a nonstandard limit control problem is obtained, and the usual methods of weak convergence for systems under heavy traffi...

A Tandem Network of Queues with Deterministic Service and Intermediate Arrivals

We present a complete analysis of delays for a tandem network of queues with deterministic service and multiple, interfering sources. The model is of a packet-based data collection system consisting of some finite, but arbitrary, number of stations connected in tandem by a unidirectional, asynchronous transmission network. Packets, or, more generally, tasks enter the system at every station, are handed from station to station in store-and-forward fashion, and exit at the downstream end; there are no intermediate departures. The stations are provided with infinite storage, and the lines between them operate concurrently and asynchronously. Intermediate sources are Poisson; the source at the head of the network can be somewhat more general than Poisson. Tasks from each source wait at each station downstream of their point of origin. We calculate the joint steady-state moment-generating functions for these waiting times, provided that the line capacities do not increase in the direction of flow; the solution contains as a special case the steady-state moment-generating function for end-to-end delay for each source.

A Two-Fold Tandem Net with Deterministic Links and Source Interference

The problem is to quantify the effect of flow interference on waiting-times in a two-fold tandem network of queues. Specifically, there are two deterministic servers (I0, I1) and two Poisson sources (S0, S1). So tasks are submitted first to Io and then to I1; S1 tasks proceed directly to I1. The two flows thus interfere at I1. Service is in order-of-arrival at both servers, and there are no defections at either server. In the case that I1 is no faster than I0, we compute the following quantities: (1) the network utilization factor; (2) the stationary moment-generating function of virtual waiting-time in I1; (3) the joint stationary moment-generating function for S0 waiting-time at each of I0, I1. The problem in the complementary case is still unsolved; a relevant difference-differential equation is presented, and an approximation technique is described.

Comparison of two packet-retransmission techniques (Corresp.)

Two methods of dealing with the overflow problem in packet switches are compared: switch-retransmission and host-retransmission. The comparison is based on approximate analytic models of tandem queueing networks and is verified by simulation.