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AbstractQueueing systems which include the possibility for a customer to return to the same server for additional service are called queueing systems with feedback. Such systems occur in computer networks for example. In these systems a chosen customer will wait in the queue, be serviced and then, with probability p, return to wait again, be serviced again and continue this process until, with probability (1 – p) = q, it departs the system never to return. The time of waiting plus service time, the nth time the customer goes through, we will call his nth sojourn time. The (random) sum of these sojourn times we will call the total sojourn time (abbreviated, sojourn time when there is no confusion which sojourn time we are talking about). In this paper we study the total sojourn time in a queueing system with feedback. We give the details for M/G/1 queues in which the decision to feedback or not is a Bernoulli process. While the details of the computations can be more difficult, the structure of the sojourn time process is unchanged for the M/G/1 queue with a more general decision process as will be shown. We assume the reader is familiar with Disney, McNickle and Simon [1].

AbstractWe present some results for M/M/1 queues with finite capacities with delayed feedback. The delay in the feedback to an M/M/1 queue is modelled as another M‐server queue with a finite capacity. The steady state probabilities for the two dimensional Markov process {N(t), M(t)} are solved when N(t) = queue length at server 1 at t and M(t) = queue length at server 2 at t. It is shown that a matrix operation can be performed to obtain the steady state probabilities. The eigenvalues of the operator and its eigenvectors are found. The problem is solved by fitting boundary conditions to the general solution and by normalizing. A sample problem is run to show that the solution methods can be programmed and meaningful results obtained numerically.

Major Speakers -- Networks of Quasi-Reversible Nodes -- The c — Server Queue with Constant Service Times and a Versatile Markovian Arrival Process -- Simulation Output Analysis for General State Space Markov Chains -- Models and Problems of Dynamic Memory Allocation -- Probabilistic Analysis of Algorithms -- Point Process Method in Queueing Theory -- Error Minimization in Decomposable Stochastic Models -- Computational Methods for Product Form Queueing Networks: Extended Abstract -- Networks of Queues, I Richard Muntz, Chairman -- Closed Multichain Product Form Queueing Networks with Large Population Sizes -- The Significance of the Decomposition and the Arrival Theorems for the Evaluation of Closed Queueing Networks -- On Computing the Stationary Probability Vector of a Network of Two Coxian Servers -- Performance and Reliability Donald Gross, Chairman -- Fitting of Software Error and Reliability Models to Field Failure Data -- Performance Evaluation of Voice/Data Queueing Systems -- Probabilistic Aspects of Simulation Peter Lewis, Chairman -- On a Spectral Approach to Simulation Run Length Control -- Generation of Some First-Order Autoregressive Markovian Sequences of Positive Random Variables with Given Marginal Distributions -- Testing for Initialization Bias in the Mean of a Simulation Output Series: Extended Abstract -- Queueing Models in Performance Analysis, I Daniel Heyman, Chairman -- Response Time Analysis for Pipelining Jobs in a Tree Network of Processors -- Mean Delays of Individual Streams into a Queue: The ?GIi/M/1 Queue -- Probabilistic Models in Performance Analysis of Computer Systems and Communication Networks Donald Gaver, Chairman -- Analysis and Design of Processor Schedules for Real Time Applications -- Modeling Real Dasd Configurations -- Bottleneck Determination in Networks of Queues -- Probabilistic Analysis of Algorithms Dave Liu, Chairman -- On the Average Difference Between the Solutions to Linear and Integer Knapsack Problems.

AbstractIn this paper we are concerned with several random processes that occur in M/G/1 queues with instantaneous feedback in which the feedback decision process is a Bernoulli process. Queue length processes embedded at various times are studied. It is shown that these do not all have the same asymptotic distribution, and that in general none of the output, input, or feedback processes is renewal. These results have implications in the application of certain decomposition results to queueing networks.