Waiting times for M/M systems under state-dependent processor sharing

We consider a system where the arrivals form a Poisson process and the required service times of the requests are exponentially distributed. The requests are served according to the state-dependent (Cohen’s generalized) processor sharing discipline, where each request in the system receives a service capacity which depends on the actual number of requests in the system. For this system we derive systems of ordinary differential equations for the LST and for the moments of the conditional waiting time of a request with given required service time as well as a stable and fast recursive algorithm for the LST of the second moment of the conditional waiting time, which in particular yields the second moment of the unconditional waiting time. Moreover, asymptotically tight upper bounds for the moments of the conditional waiting time are given. The presented numerical results for the first two moments of the sojourn times in M/M/m?PS systems show that the proposed algorithms work well.     

Optimal control for parallel queues with a single batch server

We consider a queueing system with multiple Poisson arrival queues and a single batch server that has infinite capacity and a fixed service time. The problem is to allocate the server at each moment to minimize the long-run average waiting cost. We propose a Cost-Arrival Weighted (CAW) policy for this problem based on the structure of the optimal policy of a corresponding fluid model. We show that this simple policy enjoys a superior performance by numerical experiments.     

Handling load with less stress

We study how the average performance of a system degrades as the load nears its peak capacity. We restrict our attention to the performance measures of average sojourn time and the large deviation rates of buffer overflow probabilities. We first show that for certain queueing systems, the average sojourn time of requests depends much more weakly on the load ρ than the commonly observed 1/(1−ρ) dependence for most queueing policies. For example, we show that for an M/G/1 system under the preemptive Shortest Job First (pSJF) policy, the average sojourn time varies as log (1/(1−ρ)) with load for a certain class of distributions. We observe that such results hold even for more restricted policies. We give some examples of non-preemptive policies and policies that do not use the knowledge of job sizes while scheduling, where the dependence of average sojourn time on load is significantly better than 1/(1−ρ). Similar results hold even for very simple non-preemptive threshold based policies that partition all the jobs into two job classes based on a fixed threshold and do FIFO within each class. Finally we study the large deviations rate of the queue length under a simple dedicated partition-based policy.     

The design of an efficient elevator operating system

A method is developed for finding an efficient operating policy for an office building elevator system. The method was applied to a particular eleven story building in which there were four elevator shafts. A queuing model was formulated in which the characteristics of passenger arrivals and destinations were time variable. The objective involved the minimization of the weighted sum of ratios of actual to minimum possible travel time between all pairs of floors. Simulation was used to analyze several logical routing policies for each of two methods in which demand information was used to alter the elevator route. The best policy was found to be almost twice as efficient as most of the other policies which were studied and over 25 times more efficient than another seemingly logical operating policy.     

Optimizing contracted resource capacity with two advance cancelation modes

Critical resources are often shared among different classes of customers. Capacity reservation allows each class of customers to better manage priorities of its customers but might lead to unused capacity. Unused capacity can be avoided or reduced by advance cancelation. This paper addresses the service capacity reservation for a given class of customers. The reservation process is characterized by: contracted time slots (CTS) reserved for the class of customers, requests for lengthy regular time slots (RTS) and two advance cancelation modes to cancel CTS one-period or two-period before. The optimal control under a given contract is formulated as an average cost Markov Decision Process (MDP) in order to minimize customer waiting times, unused CTS and CTS cancelation. Structural properties of optimal control policies are established via the corresponding discounted cost MDP problem. Numerical results show that two-period advance CTS cancelation can significantly improve the contract-based solution.     

Optimal control of polling models for transportation applications

We formulate and analyze a dynamic scheduling problem for a class of transportation systems in a Markov Decision Process (MDP) framework. A transportation system is represented by a polling model consisting of a number of stations and a server with switch-over costs and constraints on its movement (the model we have analyzed is intended to emulate key features of an elevator system). Customers request service in order to be transported by the server from various arrival stations to a common destination station. The objective is to minimize a cost criterion that incorporates waiting costs at the arrival stations. Two versions of the basic problem are considered and structural properties of the optimal policy in each case are derived. It is shown that optimal scheduling policies are characterized by switching functions dependent on state information consisting of queue lengths formed at the arrival stations.     

Steady state results for the M/M(a,b)/c batch-service system

The transportation system considered in this paper has a number of vehicles with capacity constraint, which take passengers from a source terminal to various destinations and return to the terminal. The trip times are considered to be independent and identically distributed random variables with a common exponential distribution. Passengers arrive at the terminal in accordance with a Poisson process. The system is operated under the following policy: when a vehicle is available and there are at least ‘a’ passengers waiting for service, then a vehicle is dispatched immediately. A recursive algorithm is derived to obtain the steady-state probability P(m, j) that there are m idle vehicles and j waiting passengers in the queue. Analytical expressions have been derived for passenger queue length distribution, average passenger queue length, the r-th moment of passenger waiting time in the queue, service batch size distribution and the average service batch size, all in terms of P(0,0).     

A comparison of outbound dispatch policies for integrated inventory and transportation decisions

This paper investigates the impact of alternative outbound dispatch policies on integrated stock replenishment and transportation decisions. The logistics literature reports that two different types of such policies are popular in current practice. These are time-based and quantity-based dispatch policies. Considering the case of stochastic demand, the paper presents analytical and numerical results showing that the cost savings obtained through quantity-based policies can be substantial. However, under a quantity-based policy, a specific delivery time cannot be quoted when the customer places an order. Hence, the paper also investigates the cost and customer waiting time implications of hybrid policies and demonstrates that hybrid policies are superior to time-based policies in terms of the resulting costs. Furthermore, although hybrid policies are not superior to quantity-based policies in terms of the resulting costs, they are superior in terms of a service measure which is quantified by the long-run average cumulative waiting time.     

Equivalence of two linear programming relaxations for broadcast scheduling

A server needs to compute a broadcast schedule for n pages whose request times are known in advance. Outputting a page satisfies all outstanding requests for the page. The goal is to minimize the average waiting time of a client. In this paper, we show the equivalence of two apparently different relaxations that have been considered for this problem.     

A New Approach to Analysis of Polling Systems

In this paper, we consider polling systems with J stations with Poisson arrivals and general service distributions attended by a cyclic server. The service discipline at each station is either exhaustive or gated. We propose a new approach to analysis of the mean waiting times in the polling systems. The outline of our method is as follows. We first define the stochastic process Q that represents an evolution of the system state, and define three types of the performance measures W _i ,H _i and F _i , which are the expected waiting times conditioned on the system state. Then from the analysis of customers at polling instants, we find their linear functional expressions. The steady state average waiting times can be derived from the performance measures by simple limiting procedures. Their actual values can be obtained by solving J(J+1) linear equations.     

Optimal admission policies for a finite queue with bursty arrivals

We consider a service system with a single server, a finite waiting room and two classes of customers with deterministic service time. Primary jobs arrive at random and are admitted whenever there is room in the system. At the beginning of each period, secondary jobs can be admitted from an infinite pool. A revenue is earned upon admission of each job, with the primary jobs bringing a higher contribution than the secondary jobs, the objective being to maximize the total discounted revenue over an infinite horizon. We model the system as a discrete time Markov decision process and show that a monotone admission policy is optimal when the number of primary arrivals has a fixed distribution. Moreover, when the primary arrival distribution varies with time according to a finite state Markov chain, we show that the optimal policy is conditionally monotone and that the numerical computation of an optimal policy, in this case, is substantially more difficult than in the case of stationary arrivals.This research was supported in part by the National Science Foundation, under grant ECS-8803061, while the author was at the University of Arizona.     

Optimal lateral transshipment policies for a two location inventory problem with multiple demand classes

We consider an inventory model for spare parts with two stockpoints, providing repairable parts for a critical component of advanced technical systems. As downtime costs for these systems are expensive, ready–for–use spare parts are kept in stock to be able to quickly respond to a breakdown of a system. We allow for lateral transshipments of parts between the stockpoints upon a demand arrival. Each stockpoint faces demands from multiple demand classes. We are interested in the optimal lateral transshipment policy. There are three ways in which a demand can by satisfied: from own stock, via a lateral transshipment, or via an emergency procedure. Using stochastic dynamic programming, we characterize and prove the structure of the optimal policy, that is, the policy for satisfying the demands which minimizes the average operating costs of the system. This optimal policy is a threshold type policy, with state-dependent thresholds at each stockpoint for every demand class. We show a partial ordering in these thresholds in the demand classes. In addition, we derive conditions under which the so-called hold back and complete pooling policies are optimal, two policies that are often assumed in the literature. Furthermore, we study several model extensions which fit in the same modeling framework.     

Optimal myopic policy for a stochastic inventory problem with fixed and proportional backorder costs

In this paper, we consider a single product, periodic review, stochastic demand inventory problem where backorders are allowed and penalized via fixed and proportional backorder costs simultaneously. Fixed backorder cost associates a one-shot penalty with stockout situations whereas proportional backorder cost corresponds to a penalty for each demanded but yet waiting to be satisfied item. We discuss the optimality of a myopic base-stock policy for the infinite horizon case. Critical number of the infinite horizon myopic policy, i.e., the base-stock level, is denoted by S. If the initial inventory is below S then the optimal policy is myopic in general, i.e., regardless of the values of model parameters and demand density. Otherwise, the sufficient condition for a myopic optimum requires some restrictions on demand density or parameter values. However, this sufficient condition is not very restrictive, in the sense that it holds immediately for Erlang demand density family. We also show that the value of S can be computed easily for the case of Erlang demand. This special case is important since most real-life demand densities with coefficient of variation not exceeding unity can well be represented by an Erlang density. Thus, the myopic policy may be considered as an approximate solution, if the exact policy is intractable. Finally, we comment on a generalization of this study for the case of phase-type demands, and identify some related research problems which utilize the results presented here.     

Social forecasting for company planning: Brian C. TWISS (Ed.) Macmillan,London, 1982, xxv + 310 pages, £25.00

The transportation system considered in this paper has a number of vehicles with no capacity constraint, which take passengers from a source terminal to various destinations and return to the terminal. The trip times are considered to be independent and identically distributed random variables with a common exponential distribution. Passengers arrive at the terminal in accordance with a Poisson process. The system is operated under the following policy: when a vehicle is available and there are at least α passengers waiting for service, then a vehicle is dispatched immediately. The passenger queue length and waiting time distributions are obtained under steady-state conditions. System performance measures such as average passenger queue length and waiting time are then derived. A minimum average cost criterion is then used to determine the optimal fleet size and dispatching policy. This is a generalization of the results of Weiss for a single-vehicle system.     

Inventory control for point-of-use locations in hospitals

Most inventory management systems at hospital departments are characterised by lost sales, periodic reviews with short lead times, and limited storage capacity. We develop two types of exact models that deal with all these characteristics. In a capacity model, the service level is maximised subject to a capacity restriction, and in a service model the required capacity is minimised subject to a service level restriction. We also formulate approximation models applicable for any lost-sales inventory system (cost objective, no lead time restrictions etc). For the capacity model, we develop a simple inventory rule to set the reorder levels and order quantities. Numerical results for this inventory rule show an average deviation of 1% from the optimal service levels. We also embed the single-item models in a multi-item system. Furthermore, we compare the performance of fixed order size replenishment policies and (R,?s,?S) policies.     

Minimizing the Total Cost in an Integrated Vendor—Managed Inventory System

In this paper we consider a complex production-distribution system, where a facility produces (or orders from an external supplier) several items which are distributed to a set of retailers by a fleet of vehicles. We consider Vendor-Managed Inventory (VMI) policies, in which the facility knows the inventory levels of the retailers and takes care of their replenishment policies. The production (or ordering) policy, the retailers replenishment policies and the transportation policy have to be determined so as to minimize the total system cost. The cost includes the fixed and variable production costs at the facility, the inventory costs at the facility and at the retailers and the transportation costs, that is the fixed costs of the vehicles and the traveling costs. We study two different types of VMI policies: The order-up-to level policy, in which the order-up-to level quantity is shipped to each retailer whenever served (i.e. the quantity delivered to each retailer is such that the maximum level of the inventory at the retailer is reached) and the fill-fill-dump policy, in which the order-up-to level quantity is shipped to all but the last retailer on each delivery route, while the quantity delivered to the last retailer is the minimum between the order-up-to level quantity and the residual transportation capacity of the vehicle. We propose two different decompositions of the problem and optimal or heuristic procedures for the solution of the subproblems. We show that, for reasonable initial values of the variables, the order in which the subproblems are solved does not influence the final solution. We will first solve the distribution subproblem and then the production subproblem. The computational results show that the fill-fill-dump policy reduces the average cost with respect to the order-up-to level policy and that one of the decompositions is more effective. Moreover, we compare the VMI policies with the more traditional Retailer-Managed Inventory (RMI) policy and show that the VMI policies significantly reduce the average cost with respect to the RMI policy.     

The relationship between preventive maintenance and manufacturing system performance

A common lament of the preventive maintenance (PM) crusaders is that production supervisors are often unwilling to lose valuable machine time when there are job waiting to be processed and do not assign high enough priority to PM. Maintenance activities that depend dynamically on system state are too complicated to implement and their overall impact on system performance, measured in terms of average tardiness or work-in-process (WIP) inventory, is difficult to predict. In this article, we present some easy to implement state-dependent PM policies that are consistent with the realities of production environment. We also develop polling models based analyses that could be used to obtain system performance metrics when such policies are implemented. We show that there are situations in which increased PM activity can lower total expected WIP (and overall tardiness) on its own, i.e., without accounting for the lower unplanned downtime. We also include examples that explain the interaction between duration of PM activity and switchover times. We identify cases in which a simple state-independent PM policy outperforms the more sophisticated state-dependent policies.     

Analysis of job assignment with batch arrivals among heterogeneous servers

We revisit the problem of job assignment to multiple heterogeneous servers in parallel. The system under consideration, however, has a few unique features. Specifically, repair jobs arrive to the queueing system in batches according to a Poisson process. In addition, servers are heterogeneous and the service time distributions of the individual servers are general. The objective is to optimally assign each job within a batch arrival to minimize the long-run average number of jobs in the entire system. We focus on the class of static assignment policies where jobs are routed to servers upon arrival according to pre-determined probabilities. We solve the model analytically and derive the structural properties of the optimal static assignment. We show that when the traffic is below a certain threshold, it is better to not assign any jobs to slower servers. As traffic increases (either due to an increase in job arrival rate or batch size), more slower servers will be utilized. We give an explicit formula for computing the threshold. Finally we compare and evaluate the performance of the static assignment policy to two dynamic policies, specifically the shortest expected completion policy and the shortest queue policy.     

Optimal parking of idle elevators under myopic and state-dependent policies

In this paper we discuss the problem of optimally parking single and multiple idle elevators under light-traffic conditions. The problem is analyzed from the point of view of the elevator owner whose objective is to minimize the expected total cost of parking and dispatching the elevator (which includes the cost incurred for waiting passengers). We first consider the case of a single elevator and analyze a (commonly used but suboptimal) state-independent myopic policy that always positions the idle elevator at the same floor. Building on the results obtained for the myopic policy, we then show that the optimal non-myopic (state-dependent) policy calls for dispatching the idle elevator to the state-dependent median of a weight distribution. Next, we consider the more difficult case of two elevators and develop an expression for the expected dispatching distance function. We show that the objective function for the myopic policy is non-convex. The non-myopic policy is found to be dependent on the state of the two idle elevators. We compute the optimal state-dependent policy for two elevators using the results developed for the myopic policy. Next, we examine the case of multiple elevators and provide a general recursive formula to find the expected dispatching distance functions. Finally, we generalize the previous models by incorporating a fixed cost for parking the idle elevators that results in a two-sided optimal policy with different regions. Every policy that we introduce and analyze is illustrated by an example. The paper concludes with a short summary and suggestions for future research.