Performance analysis of GI/M/1 queue with working vacations and vacation interruption

In this paper, we analyze a single-server vacation queue with a general arrival process. Two policies, working vacation and vacation interruption, are connected to model some practical problems. The GI/M/1 queue with such two policies is described and by the matrix analysis method, we obtain various performance measures such as mean queue length and waiting time. Finally, using some numerical examples, we present the parameter effect on the performance measures and establish the cost and profit functions to analyze the optimal service rate η during the vacation period.     

BMAP/G/1/N queue with vacations and limited service discipline

We consider a finite buffer single server queue with batch Markovian arrival process (BMAP), where server serves a limited number of customer before going for vacation(s). Single as well as multiple vacation policies are analyzed along with two possible rejection strategies: partial batch rejection and total batch rejection. We obtain queue length distributions at various epochs and some important performance measures. The Laplace–Stieltjes transforms of the actual waiting time of the first customer and an arbitrary customer in an accepted batch have also been obtained.     

Performance analysis of MAP/G/1 queue with working vacations and vacation interruption

We consider the MAP/G/1 queue with working vacations and vacation interruption. We obtain the queue length distribution with the method of supplementary variable, combined with the matrix-analytic method and censoring technique. We also obtain the system size distribution at pre-arrival epoch and the Laplace–Stieltjes transform (LST) of waiting time.     

On the optimal control of a single-server queueing system: Comment

In a recent paper by Scott and Jefferson, the optimal control of the service rate for a single-server queue with limited waiting space is treated by the maximum principle. We show that their control policies are necessarily suboptimal. Characterizations for optimal control are derived and used to obtain corresponding optimal trajectories in both nonsingular and singular regions.     

A co-evolutionary approach using information about future requests for dynamic vehicle routing problem with soft time windows

This paper presents a technique for integrating information about future customer requests to improve decision making for dynamic vehicle routing. We use a co-evolutionary approach to generate better waiting strategies such that the expected number of late-request customers who are served is maximized. An empirical evaluation of the proposed approach is performed within a previously reported hybrid genetic algorithm for the dynamic vehicle routing problem with time windows. Comparisons with other heuristic methods demonstrate the potential improvement that can be obtained through the application of the proposed approach.     

The Discrete-Time GI/Geo/1 Queue with Multiple Vacations

We study a discrete-time GI/Geo/1 queue with server vacations. In this queueing system, the server takes vacations when the system does not have any waiting customers at a service completion instant or a vacation completion instant. This type of discrete-time queueing model has potential applications in computer or telecommunication network systems. Using matrix-geometric method, we obtain the explicit expressions for the stationary distributions of queue length and waiting time and demonstrate the conditional stochastic decomposition property of the queue length and waiting time in this system.     

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.     

Queues with service times and interarrival times depending linearly and randomly upon waiting times

We consider a modification of the standardG/G/1 queue with unlimited waiting space and the first-in first-out discipline in which the service times and interarrival times depend linearly and randomly on the waiting times. In this model the waiting times satisfy a modified version of the classical Lindley recursion. We determine when the waiting-time distributions converge to a proper limit and we develop approximations for this steady-state limit, primarily by applying previous results of Vervaat [21] and Brandt [4] for the unrestricted recursionY _n+1=C _n Y _n +X _n . Particularly appealing for applications is a normal approximation for the stationary waiting time distribution in the case when the queue only rarely becomes empty. We also consider the problem of scheduling successive interarrival times at arrival epochs, with the objective of achieving nearly maximal throughput with nearly bounded waiting times, while making the interarrival time sequence relatively smooth. We identify policies depending linearly and deterministically upon the work in the system which meet these objectives reasonably well; with these policies the waiting times are approximately contained in a specified interval a specified fraction of time.     

Convergence of a queueing system in heavy traffic with general patience-time distributions

We analyze a sequence of single-server queueing systems with impatient customers in heavy traffic. Our state process is the offered waiting time, and the customer arrival process has a state dependent intensity. Service times and customer patient-times are independent; i.i.d. with general distributions subject to mild constraints. We establish the heavy traffic approximation for the scaled offered waiting time process and obtain a diffusion process as the heavy traffic limit. The drift coefficient of this limiting diffusion is influenced by the sequence of patience-time distributions in a non-linear fashion. We also establish an asymptotic relationship between the scaled version of offered waiting time and queue-length. As a consequence, we obtain the heavy traffic limit of the scaled queue-length. We introduce an infinite-horizon discounted cost functional whose running cost depends on the offered waiting time and server idle time processes. Under mild assumptions, we show that the expected value of this cost functional for the n-th system converges to that of the limiting diffusion process as n tends to infinity.     

A hybrid constraint programming approach to?the?log-truck scheduling problem

Scheduling problems in the forest industry have received significant attention in the recent years and have contributed many challenging applications for optimization technologies. This paper proposes a solution method based on constraint programming and mathematical programming for a log-truck scheduling problem. The problem consists of scheduling the transportation of logs between forest areas and woodmills, as well as routing the fleet of vehicles to satisfy these transportation requests. The objective is to minimize the total cost of the non-productive activities such as the waiting time of trucks and forest log-loaders and the empty driven distance of vehicles. We propose a constraint programming model to address the combined scheduling and routing problem and an integer programming model to deal with the optimization of deadheads. Both of these models are combined through the exchange of global constraints. Finally the whole approach is validated on real industrial data.     

Workload and Waiting Time Analyses of MAP/G/1 Queue under D-policy

We study the workload (unfinished work) and the waiting time of the queueing system with MAP arrivals under D-policy. The D-policy stipulates that the idle server begin to serve the customers only when the sum of the service times of all waiting customers exceeds some fixed threshold D. We first set up the system equations for workload and obtain the steady-state distributions of workloads at an arbitrary idle and busy points of time. We then proceed to obtain the waiting time distribution of an arbitrary customer based on the workload results. The M/G/1/D-policy queue will be investigated as a special case.     

Modeling the transplant waiting list: A queueing model with reneging

Motivated by the problem of organ allocation, we develop a queueing model with reneging that provides a stylistic representation of the transplant waiting list. The model assumes that there are several classes of patients, several classes of organs, and patient reneging due to death. We focus on randomized organ allocation policies and develop closed-form asymptotic expressions for the stationary waiting time, stationary waiting time until transplantation, and fraction of patients who receive transplantation for each patient class. Analysis of these expressions identifies the main factors that underlie the performance of the transplant waiting list and demonstrates that queueing models can prove useful when evaluating the organ allocation system. This revised version was published online in June 2006 with corrections to the Cover Date.     

The dynamical Borel-Cantelli lemma and the waiting time problems

We investigate the connection between the dynamical Borel-Cantelli and waiting time results. We prove that if a system has the dynamical Borel-Cantelli property, then the time needed to enter for the first time in a sequence of small balls scales as the inverse of the measure of the balls. Conversely if we know the waiting time behavior of a system we can prove that certain sequences of decreasing balls satisfies the Borel-Cantelli property. This allows to obtain Borel-Cantelli like results in systems like axiom A and generic interval exchanges.     

Hybrid heuristics for a short sea inventory routing problem

We consider a short sea fuel oil distribution problem where an oil company is responsible for the routing and scheduling of ships between ports such that the demand for various fuel oil products is satisfied during the planning horizon. The inventory management has to be considered at the demand side only, and the consumption rates are given and assumed to be constant within the planning horizon. The objective is to determine distribution policies that minimize the routing and operating costs, while the inventory levels are maintained within their limits. We propose an arc-load flow formulation for the problem which is tightened with valid inequalities. In order to obtain good feasible solutions for planning horizons of several months, we compare different hybridization strategies. Computational results are reported for real small-size instances.     

On the Kleinrock-Nilsson problem of optimal scheduling algorithms for time-shared systems

We seek to minimize the mean-squared deviation of a waiting time function from a desired response function over the class of waiting time functions satisfying the Kleinrock-Nilsson necessary conditions. We will characterize analytically the optimal policy as the minimum majorant in the appropriate class of the cumulative response to go. We will show that, in general, the monotonicity necessary condition results in optimal policies which depend in some sense on the future and are anticipating.Part of this paper was completed at Bell Laboratories, Naperville, Illinois.Part of this paper was completed at the University of Kentucky, Lexington, Kentucky. This author was supported by an NSF grant and the Alfred P. Sloan Foundation.     

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.     

Online-optimization of multi-elevator transport systems with reoptimization algorithms based on set-partitioning models

We develop and experimentally compare policies for the control of a system of k elevators with capacity one in a transport environment with ? floors, an idealized version of a pallet elevator system in a large distribution center of the Herlitz PBS AG in Falkensee. Each elevator in the idealized system has an individual waiting queue of infinite capacity. On each floor, requests arrive over time in global waiting queues of infinite capacity. The goal is to find a policy that, without any knowledge about future requests, assigns an elevator to each request and a schedule to each elevator so that certain expected cost functions (e.g., the average or the maximal flow times) are minimized. We show that a reoptimization policy for minimizing average squared waiting times can be implemented to run in real-time (1 s) using dynamic column generation. Moreover, in discrete event simulations with Poisson input it outperforms other commonly used policies like multi-server variants of greedy and nearest neighbor.     

The GI/M/1 queue with phase-type working vacations and vacation interruption

Consider a GI/M/1 queue with phase-type working vacations and vacation interruption where the vacation time follows a phase-type distribution. The server takes the original work at the lower rate during the vacation period. And, the server can come back to the normal working level at a service completion instant if there are customers at this instant, and not accomplish a complete vacation. From the PH renewal process theory, we obtain the transition probability matrix. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length at arrival epochs, and waiting time of an arbitrary customer. Meanwhile, we obtain the stochastic decomposition structures of the queue length and waiting time. Two numerical examples are presented lastly.     

Numerical analysis of a discrete-time finite capacity queue with a burst arrival

We study a discrete-time, multi-server, finite capacity queue with a burst arrival. Once the first job of a burst arrives at the queue, the successive jobs will arrive every time slot until the last job of the burst arrives. The number of jobs and the inter-arrival time of bursts are assumed to be generally distributed, and the service time is assumed to be equal to one slot. We propose an efficient numerical method to exactly obtain the job loss probability, the waiting time distribution and the mean queue length using an embedded Markov chain at the arrival instants of bursts.     

Assembly-like queues with finite capacity: Bounds,asymptotics and approximations

In this paper we study a queueing model of assembly-like manufacturing operations. This study was motivated by an examination of a circuit pack testing procedure in an AT & T factory. However, the model may be representative of many manufacturing assembly operations. We assume that customers fromn classes arrive according to independent Poisson processes with the same arrival rate into a single-server queueing station where the service times are exponentially distributed. The service discipline requires that service be rendered simultaneously to a group of customers consisting of exactly one member from each class. The server is idle if there are not enough customers to form a group. There is a separate waiting area for customers belonging to the same class and the size of the waiting area is the same for all classes. Customers who arrive to find the waiting area for their class full, are lost. Performance measures of interest include blocking probability, throughput, mean queue length and mean sojourn time. Since the state space for this queueing system could be large, exact answers for even reasonable values of the parameters may not be easy to obtain. We have therefore focused on two approaches. First, we find upper and lower bounds for the mean sojourn time. From these bounds we obtain the asymptotic solutions as the arrival rate (waiting room, service rate) approaches zero (infinity). Second, for moderate values of these parameters we suggest an approximate solution method. We compare the results of our approximation against simulation results and report good correspondence.