Modelling And Analysis Of Event Message Flows In Distributed Discrete Event Simulators Of Queueing Networks

Abstract

Distributed Discrete Event Simulation (DDES) has received much attention in recent years, owing to the fact that uniprocessor based serial simulations may require excessive amount of simulation time and computational resources. It is therefore natural to attempt to use multiple processors to exploit the inherent parallelism in discrete event simulations in order to speed up the simulation process. In this dissertation we study the performance of distributed simulation of queueing networks, by analysing queueing models of message flows in distributed discrete event simulators. Most of the prior work in distributed discrete event simulation can be catego­rized as either empirical studies or analytic (or formal) models. In the empirical studies, specific experiments are run on both conservative and optimistic simulators to see which strategy results in a faster simulation. There has also been increasing activity in analytic models either to better understand a single strategy or to compare two strategies. Little attention seems to have been paid to the behaviour of the interprocessor message queues in distributed discrete event simulators. To begin with, we study how to model distributed simulators of queueing networks. We view each logical process in a distributed simulation as comprising a message sequencer with associated message queues, followed by an event processor. A major contribution in this dissertation is the introduction of the maximum lookahead sequencing protocol. In maximum lookahead sequencing, the sequencer knows the time-stamp of the next message to arrive in the empty queue. Maximum lookahead is an unachievable algorithm, but is expected to yield the best throughput compared to any realisable sequencing technique. The analysis of maximum lookahead, therefore, should lead to fundamental limits on the performance of any sequencing algorithm We show that, for feed forward type simulators, with standard stochastic assump-tions for message arrival and time-stamp processes, the message queues are unstable for conservative sequencing, and for conservative sequencing with maximum lookahead and hence for optimistic resequencing, and for any resequencing algorithm that does not employ interprocessor "flow control". It follows that the resequencing problem is fundamentally unstable and some form of interprocessor flow control is necessary in order to make the message queues stable (without message loss). We obtain some generalizations of the insta­bility results to time-stamped message arrival processes with certain ergodicity properties. For feedforward type distributed simulators, we study the throughput of the event sequencer without any interprocessor flow control. We then incorporate flow control and study the throughput of the event sequencer. We analyse various flow control mechanisms. For example, we can bound the buffers of the message queues, or various logical processes can be prevented from getting too far apart in virtual time by means of a mechanism like Moving Time Windows or Bounded Lag. While such mechanisms will serve to stabilize buffers, our approach, of modelling and analysing the message flow processes in the simulator, points towards certain fundamental limits of efficiency of distributed simulation, imposed by the synchronization mechanism. Next we turn to the distributed simulation of more general queueing networks. We find an upper bound to the throughput of distributed simulators of open and closed queueing networks. The upper bound is derived by using flow balance relations in the queueing network and in the simulator, processing speed constraints, and synchronization constraints in the simulator. The upper bound is in terms of parameters of the queueing network, the simulator processor speeds, and the way the queueing network is partitioned or mapped over the simulator processors. We consider the problem of choosing a mapping that maximizes the upper bound. We then study good solutions o! this problem as possible heuristics for the problem of partitioning the queueing network over the simulator processors. We also derive a lower bound to the throughput of the distributed simulator for a simple queueing network with feedback. We then study various properties of the maximum lookahead algorithm. We show that the maximum lookahead algorithm does not deadlock. Further, since there are no syn­chronization overheads, maximum lookahead is a simple algorithm to study. We prove that maximum lookahead sequencing (though unrealisable) yields the best throughput compared to any realisable sequencing technique. These properties make maximum lookahead a very useful algorithm in the study of distributed simulators of queueing networks. To investigate the efficacy of the partitioning heuristic, we perform a study of queue­ing network simulators. Since it is important to study the benefits of distributed simula­tion, we characterise the speedup in distributed simulation, and find an upper bound to the speedup for a given mapping of the queues to the simulator processors. We simulate distributed simulation with maximum lookahead sequencing, with various mappings of the queues to the processors. We also present throughput results foT the same mappings but using a distributed simulation with the optimistic sequencing algorithm. We present a num­ber of sufficiently complex examples of queueing networks, and compare the throughputs obtained from simulations with the upper bounds obtained analytically. Finally, we study message flow processes in distributed simulators of open queueing networks with feedback. We develop and study queueing models for distributed simulators with maximum lookahead sequencing. We characterize the "external" arrival process, and the message feedback process in the simulator of a simple queueing network with feedback. We show that a certain "natural" modelling construct for the arrival process is exactly correct, whereas an "obvious" model for the feedback process is wrong; we then show how to develop the correct model. Our analysis throws light on the stability of distributed simulators of queueing networks with feedback. We show how the stability of such simulators depends on the parameters of the queueing network.