Discrete-time Service Research Articles

A discrete-time queueing network approach to performance evaluation of autonomous vehicle storage and retrieval systems

In this paper, we present a method for performance evaluation of autonomous vehicle storage and retrieval systems (AVS/RSs) with tier-captive single-aisle vehicles. A discrete-time open queueing network approach is applied. The data obtained from the evaluation of the lift and vehicle movements can be used directly as input for the general discrete service time distributions of the queueing network. Furthermore, the approach allows for the computation of the retrieval transaction time distribution as well as of the distribution of the number of transactions waiting to be stored. Consequently, not only expected values and variances but also quantiles of the performance measures can be obtained. Comparison to discrete-event simulation quantifies approximation errors resulting from the decomposition approach in the discrete-time domain. Moreover, the errors obtained by the discrete-time approach are compared to the errors obtained using a continuous-time open queueing network approach. Finally, it will be outlined how the model can be used for designing AVS/RSs according to given system requirements, such as storage capacity, throughput, height and length of the system as well as the 95% quantile of the retrieval transaction time.

Traffic flow model at fixed control signals with discrete service time distribution

Most of the models of road traffic flow at fixed-cycle controlled intersection assume stationary distributions and provide steady state results. The assumption that a constant number of vehicles can leave the system during the green phase is unrealistic in real life situations. A discrete time queuing model was developed to describe the operation of traffic flow at a road intersection with fixed-cycle signalized control and to account for the randomness in the number of vehicles that can leave the system. The results show the expected queue size in the system when the traffic is light and for abusy period, respectively. For the light period, when the traffic intensity is less than one, it takes a shorter green cycle time for vehicles to clear up than during high trafficintensity (the road junction is saturated). Increasing the number of cars that can leave the junction at the turn of the green phase reduces the number of cycle times before the queue is cleared.

Probability-Based Prediction of Degrading Dynamic Systems

This paper presents a methodology to provide the cumulative failure distribution (CDF) for degrading, uncertain, and dynamic systems. The uniqueness and novelty of the methodology is that long service time over which degradation occurs has been augmented with much shorter cycle time over which there is uncertainty in the system dynamics due to uncertain design variables. The significance of the proposed methodology is that it sets the foundation for setting realistic life-cycle management policies for dynamic systems. The methodology first replaces the implicit mechanistic model with a simple explicit meta-model with the help of design of experiments and singular value decomposition, then transforms the dynamic, time variant, probabilistic problem into a sequence of time invariant steady-state probability problems using cycle-time performance measures and discrete service time, and finally, builds the CDF as the summation of the incremental service-time failure probabilities over the planned life time. For multiple failure modes and multiple discrete service times, set theory establishes a sequence of true incremental failure regions. A practical implementation of the theory requires only two contiguous service-times. Probabilities may be evaluated by any convenient method, such as Monte Carlo and the first-order reliability method. Error analysis provides ways to control errors with regards to probability calculations and meta-model fitting. A case study of a common servo-control mechanism shows that the new methodology is sufficiently fast for design purposes and sufficiently accurate for engineering applications.

Analysis of an [formula omitted] retrial queue

We consider a single server retrial queueing system in which each customer (primary or retrial customer) has discrete service times taking on value D j with probability p j , j = 1 , 2 , … , and ∑ j = 1 ∞ p j = 1 . An arriving primary customer who finds the server busy tries later. Moreover, each retrial customer has its own orbit, and the retrial customers try to enter the service independently of each other. We call this retrial queue an M / { D n } / 1 retrial queue. A necessary and sufficient condition for this system stability is given. In the steady state, we derive the joint distribution of the state of the server and the number of customers in the retrial orbits. The explicit expressions of some performance measures are given. In addition, the steady-state distribution of the waiting time is discussed.

A single-server finite-capacity queueing system with Markov flow and discrete-time service

Concepts of a discrete Markov flow and discrete-time Markov service are defined. A single-server finite-capacity queueing system with Markov flow and discrete-time service is studied. An algorithm for computing the stationary state probability distribution of the system is designed. The main performance characteristics, such as the stationary loss probability and mean waiting time for service, are determined.

Performance comparison of some discrete service time systems

This paper analyzes and compares the slotted time operation and the non-slotted time operation of a singleserver system with deterministic service time. These two operations are commonly used to model discrete service time systems in computer or digital communications. But due to the similarity in their operations and performance, the two models may be mixed up with each other. This paper examines by means of queueing analysis the performance of a few classes of systems that differ in the service order of their customers. The means and variances of the queue lengths, waiting times and interdeparture times of FCFS systems using the slotted time operations are first obtained from their respective LST and generating function equations. These are then used for comparison with those in non-slotted time systems. The results show that although the slotted time operation in the FCFS systems can be approximated by the non-slotted time operations under heavy traffic condition, the performances under other regions and service disciplines (e.g., the LCFS and the priority systems) may deviate significantly. They must be properly adjusted if one wishes to use the simpler equations of the non-slotted time operations to approximate the slotted time operations. The comparison graphs provided in this paper supply adjustments guidelines for the careful designers.

Ill-conditioned performance functions of queueing systems

Shows that for queueing networks with deterministic or discrete service time distributions, the performance functions can be nondifferentiable at a dense subset of a given interval. The authors also show that when the service time densities are supported on small intervals, the performance function derivatives changes rapidly. The authors prove these results for a two-server cyclic network and then point out a potential generality to other queueing networks. The results indicate that the nonsmooth analysis may be useful in the area of stochastic discrete-event systems. >

The finite capacity multi-server queue with inhomogeneous arrival rate and discrete service time distribution — and its application to continuous service time problems

The finite capacity multi-server queueing model with inhomogeneous arrival rate and discrete service time distribution is developed. The system is formulated as an inhomogeneous Markov chain in discrete time. An algorithm is described to solve the model numerically. A method is then proposed for using this model to approximate the time dependent behaviour of multi-server queues with inhomogeneous arrival rate and continuous service time distribution. Empirical results are presented to show that this approximation will produce results that are accurate enough for most practical purposes.

Multi-item production control for production to order

This paper deals with the problem of production control in situations in which several types of products are produced on one machine and in which only the ordered goods can be produced. The demand is stochastic and depends on the average delivery-time. We will describe two decomposition methods: a method based on queuing theory and a method for discrete demand and discrete service-times. Both methods will be compared with a cyclic production strategy.

An approximation of known accuracy for single server queues with inhomogeneous arrival rate and continuous service time distribution

A practical method for obtaining approximate results for single server queues with inhomogeneous queues and continuous service time distribution is presented. The method is based on a discrete approximation to the continuous service time distribution. Exact results can be obtained for the corresponding queueing system with discrete service time distribution. These results are then corrected, and the likely accuracy of the corrected results is estimated. Four measures of performance are considered, idleness probability, mean and variance of number of customers in the system and virtual waiting time.

The single server queue with inhomogeneous arrival rate and discrete service time distribution

A practical method of calculating the distribution of the number of customers in the single server queueing system with inhomogeneous arrival rate and discrete service time distribution is proposed. The system is formulated as an inhomogeneous Markov chain in discrete time, leading to recurrence relations for the state probabilities. The recurrence relations are then solved numerically. Various measures of performance, such as mean and variance of the number of customers in the system and virtual waiting time can be obtained from these results. Examples are presented to demonstrate the scope of the method, including time-dependent behaviour of homogeneous queues; cyclic behaviour of queues with cyclic arrival rates; and a previously published study of an airport runway in which the author had to resort to crude interpolation to obtain results. The method can be further extended to provide a reasonably accurate approximation for some systems with continuous distributions of service times.

Message Delays in Repeated-Service Tandem Connections

One of the inherent complexities associated with queueing models for store and forward data communications networks arises from the fact that messages typically preserve their length as they traverse the system. The interarrival and service sequences at queues internal to the network are thus dependent, making standard methods of analysis realistically inappropriate (except in an approximate sense). In an effort to find methods of dealing with such nonstandard queueing systems, a model for sparsely-connected networks (or network segments) called a message channel has been studied. A message channel is defined as a tandem connection of single server queues in which the successive service times experienced by any particular customer are scaled versions of the same random variable. A number of recent results concerning the delay properties of such systems are presented. These include: ordering relations for the successive waiting times in the channel; characterizations of waiting time properties under extreme conditions (simultaneous arrival of all customers): and, simple bounds on performance parameters for systems with uniformly bounded service processes. A general integral equation for the distribution of the equilibrium cumulative waiting time in GI/(G/1)^{M} queueing systems (which are instances of uniform message channels) is also derived. The solution of this equation for systems with discrete service time distributions and Poisson arrivals is indicated and explicitly exhibited for the case of two discrete levels (corresponding to a message switching system with two packet classes).

On the busy period of discrete-time queues

This paper obtains the probability and the expectation of the length of the busy and idle periods in a discrete-time service system by means of two-component Markov chains and their first passage times. The M/G/1 model is discussed as a special case.

On the busy period of discrete-time queues

This paper obtains the probability and the expectation of the length of the busy and idle periods in a discrete-time service system by means of two-component Markov chains and their first passage times. The M/G/1 model is discussed as a special case.

Theory and numerical solution of a discrete queueing problem

Abstract A queueing problem is considered with one server, independent discrete inter‐arrival times, and independent discrete service times. The steady state distribution of the waiting time is approximated by means of the solution of certain linear equations and applying a generalization of the Lin‐Bairstow method for finding quadratic factors of polynomials.