Semi-open queuing networks: a review of stochastic models,solution methods and new research areas

Capturing the waiting times (at an external queue) for a customer to access a movable resource is an important step towards measuring customer service and system performance in manufacturing, logistics, communication and health care systems. Such waiting time measures are typically used for sizing resource and buffer capacities, and thereby minimising customer waiting time probabilities. In this regard, semi-open queuing networks (SOQNs), which decouple the arriving customers/transactions from the network resources using a synchronisation station (also known as a semaphore queue), can potentially capture the customer/transaction waiting times/costs more precisely and provide a rich network modelling construct. Hence, modelling manufacturing or service systems using SOQNs is an important step towards measuring customer flow times (sojourn times) wherein the customer waiting times at an external queue are a critical component. In this paper, we present several stochastic models for manufacturing and service systems using SOQNs and also discuss the potential applications of SOQNs. We then review the solution methods for SOQNs and also compare the numerical accuracies for three promising methods. Finally, we include the potential research areas in SOQNs.     

When should we superpose M/G/1 queues?

Conventional wisdom suggests that a fast single-server queue is preferable to a number of slower single-server queues. The rationale is that if the service rate of the superposed queue is the sum of the service rates of the individual queues, then the average waiting time for the superposed queue is smaller than the weighted average waiting time of the individual queues. This argument, however, assumes that the service time of the customers is perfectly scalable. Unfortunately, this assumption rarely holds in reality. In this paper we analyze the superposition of M/G/1 queues. We assume that there are scalable and unscalable components in the service time. On the basis of the criteria of average waiting time and average response time, we derive necessary and sufficient conditions for whether the superposition of queues is preferable. Examples of magnetic rotational disks are given to illustrate our findings.     

A Heuristic Programming Algorithm for Warehouse Location

A new heuristic programming method of solving a particular type of warehouse location problem is presented. The problem is to allocate K or less facilities to N possible locations so as to service M demand centers at minimum cost. The algorithm presented is suitable for hand calculation of medium-size problems (50 × 50) or when computerized will readily solve large-scale problems of the order of (600 × 600); i.e., 600 demand centers and 600 possible locations.     

Effects of Slow-Downs and Failure on Stochastic Service Systems

The reliability and efficiency of a stochastic service system subject to break-downs and slow-downs is investigated. A mathematical model of such a system is a single server queue with Poisson input and exponential service time. However, the server is subject to break-downs and slow-downs, the lengths of which are exponentially distributed. During these periods he serves at varying rates. Analytic expressions are derived for the mean waiting time and the mean queue length.     

Optimal Design of Multiserver Queueing Systems with Different Waiting Costs for Time in Queue and Time in Service

The optimal number of servers and service rate are characterized for multiserver systems in which waiting times in service and in queue have different costs. It is shown that the optimal choice of the service rate and the number of servers depends on the ratio of the in-service waiting cost coefficient to the in-queue waiting cost coefficient. For M/M/c systems, a single server is optimal if the cost-coefficient ratio exceeds a threshold value of one-half. For GI/M/c and M/G/c systems, the threshold ratio increases as the variation of the interarrival or service time increases.     

Performance analysis of automatic storage/retrieval systems by stochastic modelling

This study considers an automated storage/retrieval system that performs either single or dual commands and analyses it as a queuing system with two waiting spaces and one server. It is assumed that the storage and retrieval commands arrive at the system according to Poisson processes with different rates and that the service times of single and dual commands are distributed differently. The steady-state probability distribution of the number of commands in the system at the service completion epochs is derived; subsequently, the semi-Markov process is employed to obtain the distribution at an arbitrary time. Several performance measures are derived and include the expected number of commands in the system, the expected waiting time, the probability that an arbitrary command cannot enter the system (the ‘blocking probability’) and the utilization of the storage/retrieval machine.     

Energy evaluations during crack initiation

Experimental and analytical investigations of crack initation from single edge notched specimens of an amorphous polymer are reported. Experimental studies are aimed at observations of damage accumulation prior to crack initiation and the time to crack initiation under low cycle fatigue conditions. Under the particular conditions investigated here, two distinct patterns of crazing are observed: a core of crazes in front of the notch tip and peripheral, less dense crazes, which surround the core. The kinetics of the peripheral crazes resembles the behavior of short cracks observed in various materials. An abrupt change of the core craze zone is observed during a single loading cycle. This change in pattern suggests a drastic change in the stress field and is presumingly associated with the crack initiation event. For identical loading conditions the time to crack initiation is within experimental error. The analytical study involves the development of a numerical scheme for evaluation of the energy release rates due to the zone of damage prior to crack initiation. These energies are taken as the J and M integrals. The numerical scheme developed herein is based on experimentally measured craze openings and the technique of double layer potential. The results of the analysis show that while the M integral is positive and increases for both the core and the peripheral crazes, the value of J is negative and decreases for the core and increases positively for the peripheral crazes. The negative values of J are attributed to the fact that the core of crazes is embedded into an inhomogeneous stress field.     

Approximate Analysis for the Merge Configuration of an Open Queueing Network with Blocking

A merge configuration of open queueing networks with exponential service times and finite buffers is analysed. We offer an iterative algorithm to approximate the steady-state probabilities for each queue of the system. The procedure decomposes the queueing network into individual queues and analyses each individual queue in isolation. An M/M/l/N or M/G/l/N model is used for the analysis of the merging queues; an M/M/l/N with state dependent arrival rates is used for the receiving queue. The approximation method is easy to implement, requires little memory, is computationally fast and yields very accurate results.     

Action-dependent stopping times and Markov decision process with unbounded rewards

This paper presents a unifying algorithmic analysis for a general class of single server queueing systems with a state dependent Markovian input process and a phase-type service time distribution including single server queues with random and quasirandom input. Using regenerative analysis we develop numerically stable and efficient recursion schemes to compute the state probabilities. The computation of the waiting times is based on the state probabilities.     

On a Computational Approach to Some Value Functions in an M/EZ/1 Queue

An M/E_z/1 queue (i.e. a single-server queue having a Poisson arrival process and weighted-sum Erlangian service) can approximate a wide range of queueing systems quite closely. For this system a practical computational scheme is developed for determining (1) the expected amount of time the server is idle in the interval (0, t] and (2) the expected waiting time of a customer arriving at time t. The initial service load can be specified arbitrarily. Sample results and computer time requirements are also discussed.     

The buffer allocation problem for general finite buffer queueing networks

The Buffer Allocation Problem (BAP) is a difficult stochastic, integer, nonlinear programming problem. In general, the objective function and constraints of the problem are not available in a closed form. An approximation formula for predicting the optimal buffer allocation is developed based upon a two-moment approximation formula involving the expressions for M/ M/1/ K systems. The closed-form expressions of the M/ M/1/ K and M/ G/1/ K systems are utilized for the BAP in series, merge, and splitting topologies of finite buffer queueing networks. Extensive computational results demonstrate the efficacy of the approach.     

Reliability measures for dynamic multistate nonrepairable systems and their applications to system performance evaluation

The main focus of this paper is on the development of reliability measures for dynamic multistate systems which have M + 1 discrete states of working efficiency. In traditional reliability models for binary systems, one measure that is commonly used is the probability of success at some time t. For multistate systems, we can extend this definition and define the reliability of a multistate system to be the probability that the system still functions at some intermediate state k, 0 ≤ k ≤ M or higher at time t. To develop the reliability measures for multistate systems, we assume that the degradation of the multistate systems follows a Markov process and that the system can directly degrade into any lower state. The other focus of this paper is on the evaluation of multistate systems for system design. For traditional binary reliability evaluation, the better system at time t is the one that has a greater value of the area created by the integration over time of the expected value of the state of the system from time 0 to time t. This integration is also related to a customer's experience with the system over the period under consideration. For binary systems, the expected value of the state of the system at any time is equal to the probability of success at the time under consideration. Similarly, one way to evaluate multistate systems is to calculate some accumulated value which is the integration of the expected value of the state of the system from time 0 to time t. Another way to evaluate multistate systems is to estimate how much benefit the customer can receive through using the system. To obtain this measure, we include a customer's utility function over time, and develop a model that can calculate a customer's expected total utility for their experience with the system. A better system should give the customer a high total utility over time and then these measures can be used in system design and evaluation.     

Parallel time scales and two-dimensional manufacturer and individual customer warranties

We deal with a system whose failures depend on several parallel effects, such as the time in use L and the mileage H. Manufacturer warranties are typically described by a two-dimensional region in the (L, H)-plane. A proper determination of the warranty limits must be based on a two-dimensional distribution of time to failure on this plane. The aim of this paper is to demonstrate the possibility of designing individual warranties for a “nontypical” customer who has a very low or very high usage rate b = H/L, and to show a simple way to calculate warranty limits by minimizing the lifetime coefficient of variation. The latter is carried out by introducing the “best” combined time scale in the form K = (1-ε)L+εH which provides the minimal lifetime coefficient of variation.     

Fragility of superheated melts in Al–RE (Ce, Nd, Pr) alloy system

The fragility of superheated melt M, characterized by the temperature dependence of the viscosity scaled by the viscosity at the liquidus temperature, has been proposed by Bian et al. recently. In this work, the values of M of Al–RE(Ce, Nd, Pr) melts were calculated based on the viscosities measured in the range of less than 250 K above their liquidus temperatures. It was found that M has a good negative correlation with the glass-forming ability (GFA) in Al–RE binary alloy system. Although the previous study found that M could reflect the GFA in Al–Co–Ce alloy system, there is no good relationship between M and GFA in the alloy system including Al–RE and Al–Co–Ce alloys simultaneously. The relationship between fragility of superheated melts and glass-forming behavior in different Al-based alloy systems was elaborated.     

On the combined influences of Young''s modulus and stress ratio on the LEFM fatigue crack growth process: A new mechanistic approach

A new mechanistic approach (NMA) was used recently to examine the physical aspects of LEFM (long) fatigue crack growth (FCG) process in crack-ductile materials in stages I and II. In this paper, NMA is extended to examine both the physical and analytical aspects of the combined effects of Young's modulus, E and stress ratio, R, in the same stages of the same materials. It is shown that, (i) with submicroscopic cleavage or reversed shear mechanism operating in the pure form, E is the most influential intrinsic “material” property controlling FCG, (ii) E-dependence of da/dN is a natural consequence of near-crack-tip displacement control proposed previously, and (iii) the demonstrated similarity of FCG curves and the existence of characteristic “pivot points” on these curves for a “class of materials” results from E-influence which continues even at a higher R. A simple analytical model based on “strain intensity factor,” K₀, which contains E-influence implicitly and controls da/dN in all materials irrespective of class, is proposed. Model-predicted K₀-based theoretical values of threshold, “Idealised Master Growth Curves (IMGCs)” and mechanism transition point, all agreed excellently with experimental data for at least three classes of materials, i.e. steels, Al-alloys and Ti-alloys at extreme R-values of 0 and ≥ 0.6. The K₀-parameter concept is used here to raise the status of the analysis of the E-effect from a simple “normalisation” to that of direct data “representation”. Using NMA existing empirical relations are given some sound theoretical base. In addition to aiding in a clearer physical understanding of the FCG process, the unique IMGCs developed for different R-values are considered useful in quick, accurate and conservative life estimations, and performing failure analyses usually required in selection and design of materials.     

Finite waiting space bulk queueing systems

Summary This paper extends earlier work on the stationary queue length distribution of a bulk service system with finite waiting space by considering two queueing systems. The first system incorporates the feature of batch arrivals with group service; it has compound Poisson input, general service times and a single server with variable batch capacity. The second system has individually arriving customers with Erlangian interarrival time distribution, general service times and a single server with variable batch capacity.The financial support of the National Research Council of Canada, under Research Grants No. 3-644-189-60; NRC-A5639 and No. 3-641-189-10; NRC-A2796, is acknowledged.     

G-moving frames and their applications in dynamics

We consider dynamical systems (M, u) with a free action of a Lie Group G that may or may not be a symmetry group. The phase space of such systems is foliated by G-orbits isomorphic to G. A G-moving frame is identified with a family of surfaces transversal to G-orbits. Given such a G-frame, the original systems splits into two subsystems describing the motion of the G-frame and the relative motion. The paper outlines the method of G-moving frames as an adequate tool to study dynamics of the systems (M, u) with a G-action and illustrates its efficiency by examples of application to mechanical and hydrodynamical systems. Some new solutions of Euler equations for ideal fluid are found, other interesting solutions are interpreted as G-modes.     

Computer protocol and torus maps

We investigate the dynamics of maps and flows which arise from a class of models of closed queueing networks in computer science theory. The network consists of n+/ servers, one of which is a central server with a queue of size n-1. A protocol or scheduling discipline must be specified in this server to define the queueing network. The standard model gives rise to a flow on an n-torus. We consider the service protocols first in-first out (FIFO) and last in-first out (LIFO) in dimension three, for which the state spaces are modifications of a 3-torus. We present a sufficient condition on the time it takes each call to complete one cycle for the FIFO protocol which guarantees that the set of periodic orbits which involve no waiting in the queue is a global attractorfor the associated semi-flow. We also investigate the dynamics for the LIFO service protocol via a return map derived from the associated area preserving flow.     

Molecular dynamics simulations of spallation in metals and alloys

We report on the results of large-scale molecular dynamics simulations of the mechanical behavior of two-dimensional metallic systems. The specific impact phenomenon studied is that in which a flyer of mass M moving with x-velocity v impacts a target of mass 2M moving with x-velocity −v/2. Simulations of such a spallation experiment have been performed for a generic metal, modelled with an embedded atom potential and also for a Cu-Ni alloy system, modelled with truncated Lennard-Jones potentials.

Our simulations indicate cold-welding upon impact, and shock wave generation, followed by rebound from the boundaries. The alloy was less ductile than the generic metal and consequently the system came apart due to the cooperative effect of the reflected shock waves.     

A Decision-Making Model for Applications of Queueing Theory

The purpose of this article is to present and illustrate a general framework for decision-making problems associated with queueing systems. The proposed feedback model is a generalization of the type of open-loop cost model usually encountered in the queueing literature. The model is discussed in the context of simple service systems for which queueing theory can provide the required queueing statistics. The use of the model is first illustrated by an M|M| 1 example, for which solutions are obtainable in closed-form and sensitivity to parameters is easily explored. In general, extensions from the basic example will require solution of nonlinear programming problems. This is illustrated by using the constant service-time variation of the example to obtain some comparative results.