Multiclass Closed Queueing Network Research Articles

A Dynamic Model for Managing Volunteer Engagement

Non-profit organizations that provide food, shelter, and other services to people in need, rely on volunteers to deliver their services. Unlike paid labor, non-profit organizations have less control over unpaid volunteers’ schedules, efforts, and reliability. However, these organizations can invest in volunteer engagement activities to ensure a steady and adequate supply of volunteer labor. We study a key operational question of how a non-profit organization can manage its volunteer workforce capacity to ensure consistent provision of services. In particular, we formulate a multiclass closed queueing network model to characterize the optimal engagement activities for the non-profit organization to minimize the costs of enhancing volunteer engagement and reducing abandonments, while maximizing productive work done by volunteers. Because this problem appears intractable, we formulate an approximating Browning control problem in the heavy traffic limit and study the dynamic control of that system. Our solution is a nested threshold policy with explicit congestion thresholds that indicate when the non-profit should optimally pursue various types of volunteer engagement activities. A numerical example calibrated using data from a large food bank shows that our dynamic policy for deploying engagement activities can reduce the food bank's total annual cost of its volunteer operations by 31%, while lowering its throughput by only 0.01%. This improvement in performance does not require any additional resources -- it only requires that the food bank strategically deploy its engagement activities based on the number of volunteers signed up to work volunteer shifts.

Robot-storage zone assignment strategies in mobile fulfillment systems

The robotic mobile fulfillment system (MFS) is widely used for automating storage pick and pack activities in e-commerce distribution centers. In this system, the items are stored on movable storage shelves, also known as inventory pods, and brought to the order pick stations by robotic drive units. We develop stylized performance evaluation models to analyze both order picking and replenishment processes in a mobile fulfillment system storage zone, based on multi-class closed queueing network models. To analyze robot assignment strategies for multiple storage zones, we develop a two-stage stochastic model. For a single storage zone, we compare dedicated and pooled robot systems for pod retrieval and replenishment. For multiple storage zones, we also analyze the effect of assigning robots to least congested zones on system throughput in comparison to random zone assignment. The models are validated using detailed simulations. For single zones, the expected throughput time for order picking reduces to one-third of its initial value by using pooled robots instead of dedicated robots; however, the expected replenishment time estimate increases up to three times. For multiple zones, we find that robots that are assigned to storage zones with dedicated and shortest queues provide a greater throughput than robots assigned at random to the zones.

Approximation analysis of multi-class closed queueing maintenance networks with a parts inventory system and two-phase Coxian time distributions

We consider a maintenance network where a set of bases is supported by a replacement parts inventory system and a centrally located repair depot. The ordering policy for the parts is the (S, Q) inventory policy. We extended the previous results to the network, where processing times at each node follow a two-phase Coxian distribution. The proposed network was modeled as a multi-class closed queueing network with a synchronization station. To make the analysis of the network computationally tractable, we developed a two-phase approximation method. In the first phase of the method, the proposed network was analyzed with the previous algorithm based on a product-form approximation. In the second phase, a sub-network was again analyzed with the procedure of a product-form approximation method such that the state space of the sub-network was reduced. In the analysis of a sub-network, a recursive method was also used to solve balance equations by exploiting the special structure of the Markov chain. The new algorithm provided a good estimation of the performance measures of interest. In addition to being accurate, the new algorithm is simple and converges rapidly.

A multi-class closed queueing maintenance network model with a parts inventory system

The system we address is a maintenance network of repairable items where a set of bases is supported by a centrally located repair depot and a consumable replacement parts inventory system. If an item fails, a replacement part must be obtained at the parts inventory system before the failed item enters the repair depot. The ordering policy for the parts is the (S,Q) inventory policy. An approximation method for this system is developed to obtain performance measures such as steady-state probabilities of the number of items at each site and the expected backorders at the parts inventory system. The proposed system is modelled as a multi-class closed queueing network with a synchronization station and analyzed using a product-form approximation method. Particularly, the product-form approximation method is adapted so that the computational effort on estimating the parameters of the equivalent multi-class network is minimized. In analyzing a sub-network, a recursive method is used to solve balance equations by exploiting the special structure of the Markov chain. Numerical tests show that the approximation method provides fairly good estimation of the performance measures of interests.

On First-Come First-Served Versus Random Service Discipline in Multiclass Closed Queueing Networks

We consider multiclass closed queueing networks. For these networks, a lot of work has been devoted to characterizing and weakening the conditions under which a product-form solution is obtained for the steady-state distribution. From this work, it is known that, under certain conditions, all networks in which each of the stations has either the first-come first-served or the random service discipline lead to the same (product-form expressions for the) steady-state probabilities of the (aggregated) states that for each station and each job class denote the number of jobs in service and the number of jobs in the queue. As a consequence, all these situations also lead to the same throughputs for the different job classes. One of the conditions under which these equivalence results hold states that at each station all job classes must have the same exponential service time distribution. In this paper, it is shown that these equivalence results can be extended to the case with different exponential service times for jobs of different classes, if the network consists of only one single-server or multiserver station. This extension can be made despite of the fact that the network is not a product-form network anymore in that case. The proof is based on the reversibility of the Markov process that is obtained under the random service discipline. By means of a counterexample, it is shown that the extension cannot be made for closed network with two or more stations.

A multiclass closed queueing network with unconventional heavy traffic behavior

We consider a multiclass closed queueing network model analogous to the open network models of Rybko and Stolyar and of Lu and Kumar. The closed network has two single-server stations and a fixed customer population of size n. Customers are routed in cyclic fashion through four distinct classes, two of which are served at each station, and each server uses a preemptive-resume priority discipline. The service time distribution for each customer class is exponential, and attention is focused on the critical case where all four classes have the same mean service time. Letting n approach infinity, we prove a heavy traffic limit theorem that is unconventional in three regards. First, in our heavy traffic scaling of both queue-length processes and cumulative idleness processes, time is compressed by a factor of n rather than the factor of $n^2$ occurring in conventional theory. Second, the spatial scaling applied to some components of the queue-length and idleness processes is that associated with the central limit theorem, but the scaling applied to other components is that associated with the law of large numbers. Thus, in the language of queueing theory, our heavy traffic limit theorem involves a mixture of Brownian scaling and fluid scaling. Finally, the limit process that we obtain is not an ordinary reflected Brownian motion, as in conventional heavy traffic theorems, although it is related to or derived from Brownian motion.

Scheduling Networks of Queues: Heavy Traffic Analysis of a Two-Station Closed Network

We consider a multiclass closed queueing network with two single-server stations. Each class requires service at a particular station, and customers change class after service according to specified probabilities. There is a general service time distribution for each class. The problem is to schedule the two servers to maximize the long-run average throughput of the network. By assuming a large customer population and nearly balanced loading of the two stations, the scheduling problem can be approximated by a dynamic control problem involving Brownian motion. A reformulation of this control problem is solved exactly and the solution is interpreted in terms of the queueing network to obtain a scheduling rule. (We conjecture, quite naturally, that the resulting scheduling rule is asymptotically optimal under heavy traffic conditions, but no attempt is made to prove that.) The scheduling rule is a static priority policy that computes an index for each class and awards higher priority at station 1 (respectively, station 2) to classes with the smaller (respectively, larger) values of this index. An analytical comparison of this rule to any other static policy is also obtained. An example is given that illustrates the procedure and demonstrates its effectiveness.

Planning the Workpiece Populations in Flexible Manufacturing Systems by Means of Approximate Performance Analysis

The solution of a particular synthesis problem is dealt with in this paper. With reference to a multiclass closed queueing network with deterministic service times, we consider the problem of determining the populations in the network in order to achieve a fixed set of production rates. The general expression of the required workpiece populations is found by making use of the relationships characterizing the equilibrium point of an approximate performance analysis method called QNA (Queueing Network Analyzer).

The static property of a perturbed multiclass closed queueing network and decomposition

A multiclass closed queueing network is viewed as a dynamic system. It is considered that a small perturbation of an event will be propagated to other events as the system evolves. The evolution of a perturbation strongly depends on the system structure. An indecomposable network is defined as a network in which a perturbation of an event at any server will, with probability one, be finally lost or propagated to all servers in the network. It is shown that a multiclass closed queueing network may be undecomposable even if every customer cannot reach every server in the network. A necessary and sufficient condition for a multiclass network to be indecomposable is given. It is concluded that the results clearly display the dynamic feature of queueing networks. >