Multiple Customer Classes Research Articles

The Complexity of Optimal Queuing Network Control

We show that several well-known optimization problems related to the optimal control of queues are provably intractable—independently of any unproven conjecture such as P ≠ NP. In particular, we show that several versions of the problem of optimally controlling a simple network of queues with simple arrival and service distributions and multiple customer classes is complete for exponential time. This is perhaps the first such intractability result for a well-known optimization problem. We also show that the restless bandit problem (the generalization of the multi-armed bandit problem to the case in which the unselected processes are not quiescent) is complete for polynomial space.

G-networks with multiple classes of signals and positive customers

G-networks are queueing models in which the types of customers one usually deals with in queues are enriched in several ways. In Gnetworks, positive customers are those that are ordinarily found in queueing systems; they queue up and wait for service, obtain service and then leave or go to some other queue. Negative customers have the specific function of destroying ordinary or positive customers. Finally triggers simply move an ordinary customer from one queue to the other. The term “signal” is used to cover negative customers and triggers. G-networks contain these three type of entities with certain restrictions; positive customers can move from one queue to another, and they can change into negative customers or into triggers when they leave a queue. On the other hand, signals (i.e. negative customers and triggers) do not queue up for service and simply disappear after having joined a queue and having destroyed or moved a negative customer. This paper considers this class of networks with multiple classes of positive customers and of signals. We show that with appropriate assumptions on service times, service disciplines, and triggering or destruction rules on the part of signals, these networks have a product form solution, extending earlier results.

Réseaux généralisés multiclasses avec synchronisations cycliques

We present a generalized network of queues with multiple classes of customers where signals may delete customers in service according to a rooted circular list of queues. We only consider stations with processor sharing service discipline. We prove the steady-state distribution has a product form if the flow equations have a solution. These networks may be used to model synchronizations in parallel systems, losses of packets in networks, of failures with deletion of customers.

Polling systems with a patient server and state-dependent setup times

We analyze a class of cyclic service (polling) systems with multiple customer classes (stations) in which the server stops cycling upon finding the entire system empty and initiates a setup only when the polled station has at least one customer in the queue. Interest in such systems is fueled by applications in design and performance analysis of manufacturing as well as telecommunication systems, We develop a discrete Fourier transform (DFT)-based near-exact numerical technique and an approximate method for systems with any number of stations. The DFT-based algorithm is accurate but computationally demanding when either the number of stations is large or server utilization is high. In these cases, the approximate method appears to work well in a large number of numerical tests.

Polling systems with a patient server and state-dependent setup times

We analyze a class of cyclic service (polling) systems with multiple customer classes (stations) in which the server stops cycling upon finding the entire system empty and initiates a setup only when the polled station has at least one customer in the queue. Interest in such systems is fueled by applications in design and performance analysis of manufacturing as well as telecommunication systems. We develop a discrete Fourier transform (DFT)-based near-exact numerical technique and an approximate method for systems with any number of stations. The DFT-based algorithm is accurate but computationally demanding when either the number of stations is large or server utilization is high. In these cases, the approximate method appears to work well in a large number of numerical tests.

Interval Coverage in Multiclass Queues Using Batch Mean Estimates

We investigate the fixed, sample-size, batch-mean procedure for creating confidence intervals from simulated data obtained from a stochastic queueing system with multiple customer classes. We show that, for a multiclass M/M/1 queue, serial correlation between customers of the same class decreases to zero as the number of customer classes increases. We also derive a closed-form expression for the asymptotic variance of waiting time by customer type. We then empirically examine batch-mean estimator coverage for a simple queue with multiple customer classes. We find that batch-mean estimators perform better in terms of coverage and interval half width in multiclass queues, with a fixed number of observations per class, than in the traditionally studied single-class systems. We also examine the effect of multiple classes where the total computational effort remains fixed.

Blocking probabilities for multiple class batched Poisson arrivals to a shared resource

A variant of the Erlang blocking model, in which multiple classes of customers arrive as batched Poisson processes, with arbitrary batch size distributions, is investigated. This model was recently considered by Kaufman and Rege, and they showed that for the partial batch blocking discipline the state distribution is of product form. In this paper the special case of complete sharing of the resource is considered. Asymptotic approximations to the blocking probability for each class of customer are derived, when the capacity of the resource and the traffic intensities are commensurately large. The results are obtained with the help of some contour integral representations, and the use of saddle-point techniques. The main result is a uniform asymptotic approximation to the blocking probability for each class of customer. This approximation is specialized in the overloaded, critically loaded and underloaded regimes. However, the uniform approximation holds across these regimes, where the blocking probabilities may vary by several orders of magnitude. The asymptotic approximations are compared numerically with some exact results, which have appeared in the literature, for the case of single (non-batched) Poisson arrivals. It is found that the uniform approximations are very accurate, and substantially more so than the approximations for the overloaded, critically loaded and underloaded regimes. Moreover, the accuracy of the uniform asymptotic approximations to the blocking probabilities is demonstrated numerically for three examples with geometric batch size distributions for each class.

G-networks with multiple classes of negative and positive customers

In recent years a new class of queueing networks with “negative and positive” customers was introduced by one of the authors [5], and shown to have a nonstandard product form. This model has undergone several generalizations to include triggers or signals which are special forms of customers whose role is to move other customers from some queue to another queue [9, 10, 6, 7]. Positive customers are identical to the usual customers of a queueing network, while a negative customer which arrives to a queue simply destroys a positive customer. We call these generalized queueing networks G-networks. In this paper we extend the basic model of [5] to the case of multiple classes of positive customers, and multiple classes of negative customers. As in other multiple class queueing networks, a positive customer class is characterized by the routing probabilities and the service rate parameter at each service center while negative customers of different classes may have different “customer destruction” capabilities. In the present paper all service time distributions are exponential and the service centers can be of the following types: FIFO (first-in-first-out), LIFO/PR (last-in-first-out with preemption), PS (processor sharing), with class-dependent service rates.

Optimal scheduling policies in time sharing service systems

We consider optimal scheduling problems in a TSSS (Time Sharing Service System), i.e., a tandem queueing network consisting of multiple service stations, all of which are served by a single server. In each station, a customer can receive service time up to the prescribed station dependent upper bound, but he must proceed to the next station in order to receive further service. After the total amount of the received services reaches his service requirement, he departs from the network. The optimal policy for this system minimizes the long-run average expected waiting cost per unit of time over the infinite planning horizon. It is first shown that, if the distribution of customer's service requirement is DMRL (Decreasing Mean Residual Life), the policy of giving the highest priority to the customer with the most attained service time is optimal under a set of some appropriate conditions. This implies that any policy without interruptions and preemptions of services is optimal. If the service requirement is DFR (Decreasing Failure Rate), on the other hand, it is shown that the policy of giving the highest priority to the customer with the least attained service time, i.e., the so-called LAST (Least Attained Service Time first) is optimal under another set of some appropriate conditions. These results can be generalized to the case in which there exist multiple classes of customers, but each class satisfies one of the above sets of conditions.

Networks of Queues with Customers, Signals and Arbitrary Service Time Distributions

We consider a network of queues with multiple classes of customers, signals, and arbitrary service time distributions. The signals bring commands to the service nodes and may trigger customers to move instantly within the network. We consider symmetric service disciplines (e.g., processor sharing, LIFO preemptive, infinite server) similar to those in F. Kelly (Kelly, F. 1979. Reversibility and Stochastic Networks. John Wiley, New York.), and show that when each node has a single server operating under a symmetric service discipline, the stationary distribution has a product-form solution. The product form, however, depends on the service time distributions of customers at each node beyond their means. This is in contrast to the insensitivity result of the conventional Jackson networks and can be attributed to the effect of signals. When the nodes have multiple servers in parallel, the usual definition of signals does not lead to a product-form solution. We propose a new definition of signals that depends on the service effort and provides a product-form solution.

Single Facility Due Date Setting with Multiple Customer Classes

We consider the interrelated problems of (1) quoting a due date to each customer arriving to a production system modeled as a single-server queue and (2) sequencing customer orders once they are in the system. We allow several different classes of customers, each with different preferences for lead time and price. We first formulate the problem of quoting due dates under the assumption that customer orders are processed on a FCFS basis. Next, we consider the case where the firm has the option to schedule orders in other than FCFS order. For this case, we develop a heuristic for quoting due dates and sequencing orders. Simulation results suggest that policies that take into account customer price and due date preferences in scheduling and quoting due dates significantly outperform due date setting policies that do not.

Approximating queue size and waiting time distributions in general polling systems

Polling system models are extensively used to model a large variety of computer and communication networks as well as production and service systems in which multiple customer classes or a number of distinct items compete for the capacity of a common server or production facility. In this paper we describe an efficient approximation method for the steady state distributions of the queue sizes and waiting times. This method is highly accurate as demonstrated by an extensive numerical study. In addition, it is highly adaptable to a variety of arrival patterns and switching protocols, including exhaustive and gated regimes, simple cyclical systems as well as general polling tables. For a system withN stations, one finds the firstK probability density function values of the steady state queue size in any given station inO(max(N, K 2) time only. When executed on an IBM system RS/6000, we have observed an average CPU time of less than 1 second for systems with as many as 50 stations over a large variety of parameter settings.

Short run dynamics of multi-class queues

We study the dynamics of general queueing systems with multiple classes of customers undergoing self-selection. We prove that if the capacity of a queue is sufficiently large, the equilibrium arrival rate will be globally stable. As the capacity is decreased, the arrival rate typically oscillates near the equilibrium.

Open queueing networks in discrete time-some limit theorems

A finite number of nodes, each with a single server and infinite buffers, is considered in discrete time. The service may be FIFO and the service times are constant. The external arrivals and the routing decision variables form a general stationary sequence. Stability of the system is proved under these assumptions. Extension to multiple servers at a node and general stationary distributions holds. If the external input is i.i.d. and the routing is Markovian then stochastic ordering, continuity of stationary distributions, rates of convergence, a functional CLT and a functional LIL and various other limit theorems for the queue length process are also proved. Generalizations to multiple servers at nodes, customers with priority, multiple customer classes, general service length and Markov modulated external arrival cases are discussed.

A note on conservation laws for a multiclass service queueing system with setup times

This paper derives a conservation law for mean waiting times in a single-server multi-class service queueing system (MX/G/1 type queue) with setup times which may be dependent on multiple customer classes and its arrival batch size by using the work decomposition property in the queueing system with vacations.

Brownian Models of Feedforward Queueing Networks: Quasireversibility and Product Form Solutions

We consider a very general type of $d$-station open queueing network, with multiple customer classes and a more or less arbitrary service discipline at each station, but restricted by the requirement that customers always flow from lower numbered stations to higher numbered ones. To approximate the behavior of such a queueing network under heavy traffic conditions, a corresponding Brownian network model is proposed and it is shown that the approximating Brownian model reduces to a $d$-dimensional reflected Brownian motion $W$ whose state space is the nonnegative orthant. A necessary and sufficient condition for $W$ to have a product form stationary distribution (that is, a stationary distribution with independent components) and a probabilistic interpretation for that condition are given. Our interpretation involves a notion of quasireversibility analogous to that introduced by Kelly and elaborated by Walrand in their brilliant analysis of product form solutions for conventional queueing network models. Three illustrative queueing network models are discussed in detail and the analysis of these examples shows how a Brownian network approximation may have a product form stationary distribution even when the original or exact model is intractable. Particularly intriguing in that regard are two examples involving non-Poisson inputs, deterministic routing, deterministic service times and processor-sharing service disciplines.

Distribution of the workload in multiclass queueing systems with server vacations

The steady-state workload at an arbitrary time is considered for several single-server queueing systems with nonpreemptive services for multiple classes of customers (arriving according to Poisson processes) and server vacation (switchover) times. The distribution of the workload at an arbitrary point during the vacation period is obtained for systems with setup times, and for polling systems with exhaustive, gated, or globally gated service disciplines. From the stochastic decomposition property, this workload is added to the workload in the corresponding M/G/1 system without vacations to give the workload at an arbitrary time in vacation systems. Dependence of the workload distribution on the vacation parameters is studied.

Perturbation analysis of a multiclass queueing system with admission control

A service facility with multiple customer classes is considered. Each class forms an independent Poisson arrival process with admission controlled by a threshold parameter and is characterized by a general service time distribution. Gradient estimators are derived for performance measures of the system (mean system time, throughput, blocking probabilities) using perturbation analysis techniques. The approach is based on exploiting alternative sample path representations of generalized semi-Markov process models of the system, such that the resulting sample functions are continuous with respect to parameters of interest. The unbiasedness of the estimators is proved. Simulation examples are included to illustrate their properties. >

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 requireMj 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 toMj. 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 toMj, 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 forj≠i. Finally, we compare two simple control policies for this system under critical loading.

Multi-class queueing models for performance analysis of computer systems

Queueing models, networks of queues in particular, have been found especially useful for estimating the performance of computer systems. Networks of queues with multiple customer classes provide a flexible framework for modelling computer systems, where a rich set of analytical results and techniques are available. When because of the complexity of the system being modelled the analytical results cannot be applied directly, they often point to fairly accurate approximation schemes. In this paper, we present a brief survey of some of the important results and techniques from the theory of multi-class queueing networks. We also present a case study to illustrate how these results and techniques are used in a real-life situation where many of the modelling constraints are violated.