A queueing system with batch renewal input and negative arrivals

This paper studies an infinite buffer single server queueing model with exponentially distributed service times and negative arrivals. The ordinary (positive) customers arrive in batches of random size according to renewal arrival process, and joins the queue/server for service. The negative arrivals are characterized by two independent Poisson arrival processes, a negative customer which removes the positive customer undergoing service, if any, and a disaster which makes the system empty by simultaneously removing all the positive customers present in the system. Using the supplementary variable technique and difference equation method we obtain explicit formulae for the steady-state distribution of the number of positive customers in the system at pre-arrival and arbitrary epochs. Moreover, we discuss the results of some special models with or without negative arrivals along with their stability conditions. The results obtained throughout the analysis are computationally tractable as illustrated by few numerical examples. Furthermore, we discuss the impact of the negative arrivals on the performance of the system by means of some graphical representations.Comment: 12 pages, 5 Figures, conferenc

Stochastic Clearing Systems With Markovian Inputs: Performance Evaluation and Optimal Policies

This thesis studies the stochastic clearing systems which are characterized by a non-decreasing stochastic input process {Y(t), t > 0}, where Y(t) is the cumulative quantity entering the system in [0, t], and an output mechanism that intermittently and instantaneously clears the system. Examples of such systems can be found in shipment consolidation, inventory backlog, lot sizing, shuttle bus dispatch, bulk service queues, and other stochastic service and storage systems. In our model, the input process is governed by an underlying discrete-time Markov chain such that, the distribution of the input in any given period depends on the underlying state in that period. The outstanding inputs in the system are recorded in strings to keep track of the ages, i.e., the time elapsed since their arrival, of each input. The decision of when to clear the system depends on a \clearing policy" which itself depends on the input quantities, ages, and the underlying state. Clearing the system will incur a fixed cost and a variable cost depending on the quantities cleared; a penalty is charged to the outstanding inputs in every period, and such penalty is non-decreasing in both the quantities and the ages of the inputs. We model the system as a tree structured Markov chain with Markovian input processes and evaluate the clearing policies with respect to the expected total costs over a finite horizon, the expected total discounted cost over an infinite horizon, as well as the expected average total cost per period over an infinite horizon. Relying on theories of Markov Decision Processes and stochastic dynamic programming, we then proceed to show some properties unique to the optimal clearing policies, and prove that a state-dependent threshold policy can be optimal under special conditions. We develop algorithms or heuristics to evaluate a given clearing policy and find the optimal clearing policy. We also use Matrix Analytic Methods to evaluate a given clearing policy and develop an efficient heuristic to find near-optimal clearing policies. Finally, we conduct extensive numerical analyses to verify the correctness, complexity, and optimality gap of our algorithms and heuristics. Our numerical examples successfully demonstrate the analytical results we proved

Service systems with balking based on queueing time

We consider service systems with balking based on queueing time, also called queues with wait-based balking. An arriving customer joins the queue and stays until served if and only if the queueing time is no more than some pre-specified threshold at the time of arrival. We assume that the arrival process is a Poisson process. We begin with the study of theM/G/1 system with a deterministic balking threshold. We use level-crossing argument to derive an integral equation for the steady state virtual queueing time (vqt) distribution. We describe a procedure to solve the equation for general distributions and we solve the equation explicitly for several special cases of service time distributions, such as phase type, Erlang, exponential and deterministic service times. We give formulas for several performance criteria of general interest, including average queueing time and balking rate. We illustrate the results with numerical examples. We then consider the first passage time problem in an M/PH/1 setting. We use a fluid model where the buffer content changes at a rate determined by an external stochastic process with finite state space. We derive systems of first-order linear differential equations for both the mean and LST (Laplace-Stieltjes Transform) of the busy period in the fluid model and solve them explicitly. We obtain the mean and LST of the busy period in the M/PH/1 queue with wait-based balking as a special limiting case of the fluid model. We illustrate the results with numerical examples. Finally we extend the method used in the single server case to multi-server case. We consider the vqt process in an M/G/s queue with wait-based balking. We construct a single server system, analyze its operating characteristics, and use it to approximate the multi-server system. The approximation is exact for the M/M/s and M/G/1 system. We give both analytical results and numerical examples. We conduct simulation to assess the accuracy of the approximation

Stochastic Clearing Models with Applications in Shipment Consolidation

This dissertation focuses on the average cost and service performance models in the shipment consolidation setting, which is treated as an application of stochastic clearing models. Specifically, we consider generalized control policies, generalized demand pattern, multi-item systems, and alternative performance criteria, where various techniques in stochastic analysis and stochastic optimal control are applied. By using stochastic impulsive control technique, we prove that, in the single item shipment consolidation model with drifted Brownian motion demand, the optimal quantity-based policy achieves the least average cost in the long run, among the admissible policies. In multi-item shipment consolidation model, we propose a (Q+τ ) policy and an instantaneous rate policy. We prove that among all (Q + τ ) policies, either a quantity-based policy or a time-based policy is optimal in terms of average cost. Furthermore, we demonstrate that the optimal instantaneous rate policy would dominate the optimal (Q + τ ) policy in terms of average cost. In terms of service performance criteria, we propose average order delay in the single-item case and average weighted delay rate in the multi-item case. From a martingale point of view, we provide a unified method to calculate the service measures. Moreover, by revealing new properties of truncated random variables, we provide comparative results among different control policies in terms of the service measures. Finally, we provide an analytical integrated inventory/hybrid consolidation model, and give comparative results in the integrated inventory/shipment consolidation models in terms of service measures and average cost