Dynamic server allocation for unstable queueing networks with flexible servers

This paper is concerned with the dynamic assignment of servers to tasks in queueing networks where demand may exceed the capacity for service. The objective is to maximize the system throughput. We use fluid limit analysis to show that several quantities of interest, namely the maximum possible throughput, the maximum throughput for a given arrival rate, the minimum arrival rate that will yield a desired feasible throughput, and the optimal allocations of servers to classes for a given arrival rate and desired throughput, can be computed by solving linear programming problems. We develop generalized round-robin policies for assigning servers to classes for a given arrival rate and desired throughput, and show that our policies achieve the desired throughput as long as this throughput is feasible for the arrival rate. We conclude with numerical examples that illustrate the points discussed and provide insights into the system behavior when the arrival rate deviates from the one the system is designed for.     

On stationary tandem queueing networks with job feedback

The class of tandem queueing networks with job feedback is studied under stationarity conditions on the arrival and service times sequences. Each job, after completing service in the last queue, is fed back (rerouted) to the first one, a random number of times, before leaving the system. The average execution time per job is exactly computed, as the number of jobs becomes large, and is minimized under mild conditions. The degree of parallelism achieved in the processing is also computed. The issue of rate-stability of the system is then considered. The network is defined to be rate-stable iff the job departure rate is equal to the job arrival rate; that depends heavily on the dynamic feedback policy we employ to place rerouted jobs in specific places of the front queue buffer of the network. The condition under which the network is rate-stable is specified, and a dynamic feedback policy is constructed, which rate-stabilizes the system under the maximum possible job arrival rate; thus, it maximizes the dynamic throughput of the network. Other related results concerning the performance of tandem networks with feedback are obtained.Research supported in part by grants NSF-DDM-RIA-9010778, NSF-NCR-9116268, NSF-NCR-NYI-9258 507, by an AT&T Foundation grant and a GTE Fellowship.     

Analysis of a Fork/Join Synchronization Station with Inputs from Coxian Servers in a Closed Queuing Network

Fork/join stations are commonly used to model the synchronization constraints in queuing models of computer networks, fabrication/assembly systems and material control strategies for manufacturing systems. This paper presents an exact analysis of a fork/join station in a closed queuing network with inputs from servers with two-phase Coxian service distributions, which models a wide range of variability in the input processes. The underlying queue length and departure processes are analyzed to determine performance measures such as throughput, distributions of the queue length and inter-departure times from the fork/join station. The results show that, for certain parameter settings, variability in the arrival processes has a significant impact on system performance. The model is also used to study the sensitivity of performance measures such as throughput, mean queue lengths, and variability of inter-departure times for a wide range of input parameters and network populations.     

A statistical Markov chain approximation of transient hospital inpatient inventory

Inventory levels are critical to the operations, management, and capacity decisions of inventory systems but can be difficult to model in heterogeneous, non-stationary throughput systems. The inpatient hospital is a complicated throughput system and, like most inventory systems, hospitals dynamically make managerial decisions based on short term subjective demand predictions. Specifically, short term hospital staffing, resource capacity, and finance decisions are made according to hospital inpatient inventory predictions. Inpatient inventory systems have non-stationary patient arrival and service processes. Previously developed models present poor inventory predictions due to model subjectivity, high model complexity, solely expected value predictions, and assumed stationary arrival and service processes. Also, no models present statistical testing for model significance and quality-of-fit. This paper presents a Markov chain probability model that uses maximum likelihood regression to predict the expectations and discrete distributions of transient inpatient inventories. The approach has a foundation in throughput theory, has low model complexity, and provides statistical significance and quality-of-fit tests unique to this Markov chain. The Markov chain is shown to have superior predictability over Seasonal ARIMA models.     

A critically loaded multiclass Erlang loss system

We consider a generalization of the classical Erlang loss model to multiple classes of customers. Each of the J customer classes has an associated Poisson arrival process and an arbitrary holding time distribution. Classj customers requireM _j servers simultaneously. We determine the asymptotic form of the blocking probabilities for all customer classes in the regime known as critical loading, where both the number of servers and offered load are large and almost equal. Asymptotically, the blocking probability of classj customers is proportional toM _j . This asymptotic result provides an approximation for the blocking probabilities which is reasonably accurate. We also consider the behavior of the Erlang fixed point (reduced load approximation) for this model under critical loading. Although the blocking probability of classj customers given by the Erlang fixed point is again asymptotically proportional toM _j , the Erlang fixed point typically gives the wrong limit. Next we show that under critical loading the throughputs have a pleasingly simple form of monotonicity with respect to arrival rates: the throughput of classi is increasing in the arrival rate of classi and decreasing in the arrival rate of classj forji. Finally, we compare two simple control policies for this system under critical loading.     

Optimal control of a removable and non-reliable server in an infinite and a finite M/H2/1 queueing system

This paper deals with a single removable and non-reliable server in both an infinite and a finite queueing system with Poisson arrivals and two-type hyper-exponential distribution for the service times. The server may be turned on at arrival epochs or off at service completion epochs. Breakdown and repair times of the server are assumed to follow a negative exponential distribution. Conditions for a stable queueing system, that is steady-state, are provided. Cost models for both system capacities are respectively developed to determine the optimal operating policy numerically at minimum cost. This paper provides the minimum expected cost and the optimal operating policy based on assumed numerical values given to the system parameters, as well as to the cost elements. Sensitivity analysis is also investigated.     

Optimization of transit route network,vehicle headways and timetables for large-scale transit networks

This paper presents a metaheuristic method for optimizing transit networks, including route network design, vehicle headway, and timetable assignment. Given information on transit demand, the street network of the transit service area, and total fleet size, the goal is to identify a transit network that minimizes a passenger cost function. Transit network optimization is a complex combinatorial problem due to huge search spaces of route network, vehicle headways, and timetables. The methodology described in this paper includes a representation of transit network variable search spaces (route network, headway, and timetable); a user cost function based on passenger random arrival times, route network, vehicle headways, and timetables; and a metaheuristic search scheme that combines simulated annealing, tabu, and greedy search methods. This methodology has been tested with problems reported in the existing literature, and applied to a large-scale realistic network optimization problem. The results show that the methodology is capable of producing improved solutions to large-scale transit network design problems in reasonable amounts of time and computing resources.     

Capacity planning in networks of queues with manufacturing applications

This paper addresses capacity planning in systems that can be modeled as a network of queues. More specifically, we present an optimization model and solution methods for the minimum cost selection of capacity at each node in the network such that a set of system performance constraints is satisfied. Capacity is controlled through the mean service rate at each node. To illustrate the approach and how queueing theory can be used to measure system performance, we discuss a manufacturing model that includes upper limits on product throughput times and work-in-process in the system. Methods for solving capacity planning problems with continuous and discrete capacity options are discussed. We focus primarily on the discrete case with a concave cost function, allowing fixed charges and costs exhibiting economies of scale with respect to capacity to be handled.     

Novel travel cost functions based on morning peak commuting equilibrium

This paper derives two novel travel cost functions by formulating a morning commuting equilibrium model that incorporates traffic congestion based on fundamental traffic flow diagram. The travel cost functions have components to represent traversal cost, waiting queuing costs and early arrival penalty. It is found that the equilibrium travel cost is a concave function of the total demand in the uncongested regime but an increasing linear function of the total demand in the congested regime.     

Fractional Brownian Heavy Traffic Approximations of Multiclass Feedforward Queueing Networks

We consider multiclass feedforward queueing networks under first in first out and priority service disciplines driven by long-range dependent arrival and service time processes. We show that in critical loading the normalized workload, queue length and sojourn time processes can converge to a multi-dimensional reflected fractional Brownian motion. This weak heavy traffic approximation is deduced from a deterministic pathwise approximation of the network behavior close to constant critical load in terms of the solution of a Skorokhod problem. Since we model the doubly infinite time interval, our results directly cover the stationary case.AMS subject classification: primary 90B15, secondary 60K25, 68M20     

Stochastic models of space priority mechanisms with Markovian arrival processes

In this paper, we study a single-server queue with finite capacity in which several space priority mechanisms are implemented. The arrival process is the general Markovian arrival process (MAP) which has been used to model the bursty arrival processes commonly arising in communication applications. The service times are generally distributed. These buffer mechanisms enable the Asynchronous Transfer Mode (ATM) layer to adapt the quality of the cell transfer to the quality of service requirements of the specific broadband ISDN services and to improve the utilization of the network resources. This is done by a selective discarding of cells according to the class they belong to. Computable expressions for various performance parameters are obtained. Numerical results are given for the case of a two-state Markov-modulated Poisson process (MMPP) and deterministic service times. The values derived can be used to evaluate the benefits of using priorities in an ATM network when the traffic is bursty and to make a comparative study of the buffer mechanisms. These results extend the models previously developed, which were limited to Poisson arrivals.     

Equivalent single-queue–single-server model for a Pentium processor

In this paper, a Pentium processor is represented as a queuing network. The objective of this paper is to deduce an equivalent single-queue–single-server model for the original queuing network. Closed-form expressions for the equivalent service rate, equivalent queue lengths, equivalent response and waiting times of the equivalent single-queue–single-server model are derived and plotted. For large values of arrival rate, queue lengths increase faster than the response times and waiting times for both the cases. Performance measures like, queue lengths, response times and waiting times are higher for lower service rates and lower for higher service rates (which is expected) of the different servers in the original queuing network. Also, the reliability in estimating performance measures for homogeneous workloads is much better than that for heterogeneous workloads.     

Throughput,flow times,and service level in an unreliable assembly system

This paper considers an unreliable assembly network where different types of components are processed by two separate work centers before being merged at an assembly station. The operation complexity of the system is a result of finite inter-station buffers, uncertain service times, and random breakdowns that lead to blocking at the work centers and starvation at the assembly station. The objective of this study is to gain an understanding of the behavior of such systems so that we can find a way to maximize the system throughput while maintaining the required customer service level. By constructing appropriate Markov processes, we obtain the probability distribution of the production flow time and derive formulas for throughput, the loss probability of type-2 workpieces, and the mean flow time. We present expressions for average work-in-process (WIP) and study their monotone properties. Using the distribution of the flow time, a customer service level can be defined and computed. We then formulate a system optimization model that can be used to maximize the throughput while maintaining an acceptable service level.     

Optimizing flow rates in a queueing network with side constraints

In this paper, modified versions of the classical deterministic maximum flow and minimum cost network flow problems are presented in a stochastic queueing environment. In the maximum flow network model, the throughput rate in the network is maximized such that for each arc of the network the resulting probability of finding congestion along that arc in excess of a desirable threshold does not exceed an acceptable value. In the minimum cost network flow model, the minimum cost routing of a flow of given magnitude is determined under the same type of constraints on the arcs. After proper transformations, these models are solved by Ford and Fulkerson's labeling algorithm and out-of-kilter algorithm, respectively.     

Multi-objective time–cost trade-off in dynamic PERT networks using an interactive approach

We develop a multi-objective model for the time–cost trade-off problem in a dynamic PERT network using an interactive approach. The activity durations are exponentially distributed random variables and the new projects are generated according to a renewal process and share the same facilities. Thus, these projects cannot be analyzed independently. This dynamic PERT network is represented as a network of queues, where the service times represent the durations of the corresponding activities and the arrival stream to each node follows a renewal process. At the first stage, we transform the dynamic PERT network into a proper stochastic network and then compute the project completion time distribution by constructing a continuous-time Markov chain. At the second stage, the time–cost trade-off problem is formulated as a multi-objective optimal control problem that involves four conflicting objective functions. Then, the STEM method is used to solve a discrete-time approximation of the original problem. Finally, the proposed methodology is extended to the generalized Erlang activity durations.     

Maximum-throughput dynamic network flows

This paper presents and solves the maximum throughput dynamic network flow problem, an infinite horizon integer programming problem which involves network flows evolving over time. The model is a finite network in which the flow on each arc not only has an associated upper and lower bound but also an associated transit time. Flow is to be sent through the network in each period so as to satisfy the upper and lower bounds and conservation of flow at each node from some fixed period on. The objective is to maximize the throughput, the net flow circulating in the network in a given period, and this throughput is shown to be the same in each period. We demonstrate that among those flows with maximum throughput there is a flow which repeats every period. Moreover, a duality result shows the maximum throughput equals the minimum capacity of an appropriately defined cut. A special case of the maximum dynamic network flow problem is the problem of minimizing the number of vehicles to meet a fixed periodic schedule. Moreover, the elegantsolution derived by Ford and Fulkerson for the finite horizon maximum dynamic flow problem may be viewed as a special case of the infinite horizon maximum dynamic flow problem and the optimality of solutions which repeat every period.     

Optimal control of an M/E<Subscript><Emphasis Type="Italic">k</Emphasis></Subscript>/1 queueing system with removable service station subject to breakdowns

In this paper we deal with a single removable service station queueing system with Poisson arrivals and Erlang distribution service times. The service station can be turned on at arrival epochs or off at departure epochs. While the service station is working, it is subject to breakdowns according to a Poisson process. When the station breaks down, it requires repair at a repair facility, where the repair times follow the negative exponential distribution. Conditions for a stable queueing system, that is steady-state, are provided. The steady-state results are derived and it is shown that the probability that the service station is busy is equal to the traffic intensity. Following the construction of the total expected cost function per unit time, we determine the optimal operating policy at minimum cost.     

Optimal Release Times in Single-Stage Manufacturing Systems with Blocking: Optimal Control Perspective

This paper concerns a due-date matching problem in a single-stage manufacturing system. Given a finite sequence of jobs and their service order, and given the delivery due date of each job, the problem is to choose the jobs release (arrival) times so as to match as closely as possible their completion times to their respective due dates. The system is modelled as a deterministic single-server FIFO queue with an output buffer for storing jobs whose service is completed prior to their due dates. The output buffer has a finite capacity; when it is full, the server is being blocked. Associated with each job there is a convex cost function penalizing its earliness as well as tardiness. The due-date matching problem is cast as an optimal control problem, whose objective is to minimize the sum of the above cost functions by the choice of the jobs arrival (release) times. Time-box upper-bound and lower-bound constraints are imposed on the jobs output (delivery) times. The optimal-control setting brings to bear on the development of fast and efficient algorithms having intuitive geometric appeal and potential for online implementation.Communicated by W. B. GongResearch supported in part by the National Science Foundation under Grant ECS-9979693 and by the Georgia Tech Manufacturing Research Center under Grant B01-D06.     

Service network design with management and coordination of multiple fleets

We present a new optimization model for the tactical design of scheduled service networks for transportation systems where several entities provide service and internal exchanges and coordination with neighboring systems is critical. Internal exchanges represent border crossings necessitating changes of vehicles, while the coordination with neighboring systems represents intermodal operations. For a given demand, the model determines departure times of the services such that throughput time of the demand in the system is minimized. The model is an extension of the design-balanced capacitated multicommodity network design model that we denote service network design with asset management and multiple fleet coordination to emphasize the explicit modeling of different vehicle fleets. Data from a real-world problem addressing the planning of new rail freight services across borders serves to illustrate the capabilities of the formulation. We analyze how synchronization with collaborating services and removal of border-crossing operations impact the throughput time for the freight. We identify a significant potential for system performance enhancement from synchronization among collaborating services for the problem studied.