Morton J. M. Posner

Level Crossings in Point Processes Applied to Queues: Single-Server Case

This paper introduces a new methodology for obtaining the stationary waiting time distribution in single-server queues with Poisson arrivals. The basis of the method is the observation that the stationary density of the virtual waiting time can be interpreted as the long-run average rate of downcrossings of a level in a stochastic point process. Equating the total long-run average rates of downcrossings and upcrossings of a level then yields an integral equation for the waiting time density function, which is usually both a linear Volterra and a renewal-type integral equation. A technique for deriving and solving such equations is illustrated by means of detailed examples.

The System Point Method in Exponential Queues: A Level Crossing Approach

The purpose of this paper is to describe the new System Point method for analyzing queues. It considers the stationary probability distribution of the waiting time in variations of the M/M/R queue with first come first served discipline for a large class of service mechanisms. It is shown that the stationary probability density function of the waiting time evaluated at w > 0 can be interpreted as the long run average of the number of times that the virtual wait becomes less than w, per unit time. Theorems are presented which establish this interpretation of the probability density function of the virtual waiting time in terms of point processes generated by level crossings in the state space. These theorems, in combination with the principle of stationary set balance, generate a system of model equations that can be written directly. In addition, the forms of these equations are often linear Volterra integral equations of the second kind with parameter, which yield direct analytical solutions. An analogous theorem is proved for a variant of M/G/1. Two illustrative examples are presented.

Analysis of a production-inventory system with unreliable production facility

The general purpose of this research is to incorporate the element of machine imperfection into production-inventory models. Most existing production-inventory models simply ignore the reliability factor, thereby tacitly assuming perfectly reliable systems. Machine failures, however, do occur and could be very disruptive-particularly in a highly automated production environment. In the model studied in this work, we consider a relatively simplified production situation: one machine, constant production rate, and a compound Poisson demand process for the product; but we superimpose the reliability feature comprising the machine failure process and the ensuing repair actions. For this situation we derive an explicit closed form solution for the steady-state distribution of the inventory level, and this result is then used to compute system performance indices of interest related to service level to customers and machine utilization. Some numerical results demonstrate the effect of the reliability parameters on these performance indices.

An approximation method for the M/G/1 retrial queue with general retrial times

Abstract In this paper, we consider a single-server queueing system with no waiting space. Customers arrive according to a Poisson process with rate λ. An arriving customer receives immediate service if he finds the server idle; otherwise he will retry for service after a certain amount of time. Blocked customers will repeatedly retry for service until they get served. The times between consecutive attempts are assumed to be independently distributed with common distribution function T (·). The service times of customers are drawn from a common distribution function B (·). We show that the number of customers in the system in steady-state can be decomposed into two independent random variables: the number of customers in the corresponding ordinary M/G/1 queue (with unlimited waiting space) and the number of customers in the retrial queue given that the server is idle. Applying this decomposition property, an approximation method for the calculation of the steady-state queue size distribution is proposed and some properties of the approximation are discussed. It is demonstrated through numerical results that the approximation works very well for models of practical interest.

A two-stage tandem queue attended by a moving server with holding and switching costs

We consider a two-stage tandem queue attended by a moving server, with homogeneous Poisson arrivals and general service times. Two different holding costs for stages 1 and 2 and different switching costs from one stage to the other are considered. We show that the optimal policy in the second stage is greedy; and if the holding cost rate in the second stage is greater or equal to the rate in the first stage, then the optimal policy in the second stage is also exhaustive. Then, the optimality condition for sequential service policy in systems with zero switchover times is introduced. Considering some properties of the optimal policy, we then define a Triple-Threshold (TT) policy to approximate the optimal policy in the first stage. Finally, a model is introduced to find the optimal TT policy, and using numerical results, it is shown that the TT policy accurately approximates the optimal policy.

Single-Server Queues with Service Time Dependent on Waiting Time

This paper discusses a class of queuing models in which the service time of a customer at a single-server facility is dependent on the amount of time that customer has spent queuing for service. A complete investigation is carried out in the particular case of exponential service times with the parameter having a step-function dependency on the waiting time. The stationary waiting-time distribution is investigated for the queue discipline by which customers are served in order of arrival. This class of model is particularly suited for applications wherein customer or item deterioration occurs during waiting or where the server wishes to provide a more appropriate level of service to counter the negative effects of customer impatience and loss of good will.

A continuous-review inventory system with lost sales and variable lead time

Using a system-point (SP) method of level crossings, we derive the stationary distribution of the inventory level (stock on hand) in a continuous-review inventory system with compound Poisson demand, Erlang as well as hyperexponentially distributed lead times, and lost sales. This distribution is then used to formulate long-run average cost functions with/without a service level constraint. Some numerical results are also presented, and compared with the Hadley and Whitin heuristic.

A correlated M/G/1-type queue with randomized server repair and maintenance modes

We consider a single machine, subject to breakdown, that produces items to inventory, continuously and uniformly. While the machine is working (an ON period), there is a deterministic net flow into the buffer. There are also two different types of OFF periods; one is a repair operation initiated after a breakdown, and the other is a preventive maintenance operation. The length of an OFF period depends on the length of the preceding ON period. During OFF periods there is a uniform outflow from the inventory buffer. The buffer content process can then be described as a fluid model. The main tool employed in determining its steady-state law exploits an equivalence relationship between the buffer content process and the virtual waiting time process of a special single-server queue in which the interarrival times and the service requirements depend on each other. We present an appropriate cost function and provide an optimization scheme illustrated by some numerical results.

Closed Finite Queuing Networks with Time Lags and with Several Classes of Units

The authors have previously investigated a broad class of closed, finite queuing networks with time lags, in which all units were considered to be identical. In this paper, that work is extended to the treatment of systems having several classes of units in which units belonging to different classes may have different sets of service parameters at the stations, different sets of travel-time distributions between stations, and different routing procedures through the system.

Sole versus dual sourcing in a continuous-review inventory system with lost sales

While existing cost analysis studies on the problem of sole versus dual sourcing only deal with the backorder case, we present an exact treatment of this problem in the context of a lost sales inventory system with compound Poisson demand and exponentially distributed lead times. The stationary distribution of the inventory level (stock on hand) and the corresponding cost functions with/without a service level constraint are derived using a level-crossing methodology, and numerical results are presented.