Two-class Priority Research Articles

A stochastic fluid model approach to the stationary distribution of the maximal priority process

We consider a two-class priority queue in which the priority of a customer increases linearly at some constant, class-dependent rate. We describe the related maximum priority process { ( M 1 ( t ) , M 2 ( t ) ) : t ≥ 0 } , which is of interest since the stationary distribution of the maximum priority process at the times of the commencement of service gives information about the waiting time distributions of the two classes of customers. In the case where service times are exponential, we map a two-class maximum priority process { ( M 1 ( t ) , M 2 ( t ) ) : t ≥ 0 } to a tandem fluid queue, and use this mapping in order to derive the stationary distribution of { ( M 1 ( t ) , M 2 ( t ) ) : t ≥ 0 } . Further, we extended these results to the maximum priority process in which service times are phase-type distributed. We illustrate the theory with numerical examples.

A Fluid-Diffusion-Hybrid Limiting Approximation for Priority Systems with Fast and Slow Customers

The paper “Fluid-Diffusion-Hybrid (FDH) Approximation” proposes a new heavy-traffic asymptotic regime for a two-class priority system in which the high-priority customers require substantially larger service times than the low-priority customers. In the FDH limit, the high-priority queue is a diffusion, whereas the low-priority queue operates as a (random) fluid limit, whose dynamics are driven by the former diffusion. A characterizing property of our limit process is that, unlike other asymptotic regimes, a non-negligible proportion of the customers from both classes must wait for service. This property allows us to study the costs and benefits of de-pooling, and prove that a two-pool system is often the asymptotically optimal design of the system.

SRPT Scheduling Discipline in Many-Server Queues with Impatient Customers

The shortest-remaining-processing-time (SRPT) scheduling policy has been extensively studied, for more than 50 years, in single-server queues with infinitely patient jobs. Yet, much less is known about its performance in multiserver queues. In this paper, we present the first theoretical analysis of SRPT in multiserver queues with abandonment. In particular, we consider the [Formula: see text] queue and demonstrate that, in the many-sever overloaded regime, performance in the SRPT queue is equivalent, asymptotically in steady state, to a preemptive two-class priority queue where customers with short service times (below a threshold) are served without wait, and customers with long service times (above a threshold) eventually abandon without service. We prove that the SRPT discipline maximizes, asymptotically, the system throughput, among all scheduling disciplines. We also compare the performance of the SRPT policy to blind policies and study the effects of the patience-time and service-time distributions. This paper was accepted by Baris Ata, stochastic models & simulation.

Delay analysis of a two-class priority queue with external arrivals and correlated arrivals from another node

Calculation of the delay distribution of (low-priority) customers in priority queues is difficult, especially when the arrival process is correlated. In this paper, we find the generating functions of the delays of high- and low-priority customers in a discrete-time queue where the arrivals are generated by a superposition of two processes, an independent one and a correlated one that can bring an extra customer in the buffer during a slot. The latter process can be seen as a simplified model for the output process of another priority queue. As a consequence, our results can be used to estimate the delay in the second stage of a tandem queueing system with an extra exogenous arrival process. A tandem priority queue is not of product-form type and its analysis is known to be a hard problem. This paper can lead to a first approximate analysis that can later be extended to incorporate the correlation in the output process of a priority queue more accurately.

Asymptotics of queue length distributions in priority retrial queues

We calculate asymptotics of the distribution of the number of customers in orbit in a two-class priority retrial M∕G∕1-type queueing model. In this model, priority customers wait in line while non-priority customers join an orbit and retry later. Although the generating function and moments of the number of customers in orbit have been analyzed before, asymptotics of the distribution have not been thoroughly investigated, mainly because of the complex nature of the generating function. We show that we can use singularity analysis of the probability generating function to do just that. Our results show that different regimes exist for these asymptotics in case of light-tailed service times: in what we call the ‘priority regime’, the tail asymptotics have the same decay (∼cn−3∕2R−n) as in the priority non-retrial queue and the retrial rate only influences the constant c. In the ‘retrial regime’, the retrial rate also influences the sub-exponential factor of the asymptotics. In this regime, asymptotics are very similar to asymptotics in retrial queues without (priority) waiting line. Finally, we also analyze the case that the service time distribution is power law (with or without exponential cut-off) using the same technique.

Performance analysis of service systems with priority upgrades

In this paper, we study the performance of service systems with priority upgrades. We model the service system as a single-server two-class priority queue, with queue 1 as the normal queue and queue 2 as the priority queue. The queueing model of interest has various applications in healthcare services, perishable inventory and project management. We comprehensively examine the system’s stationary distribution, computational algorithm design and sensitivity analysis. We observe that when queue 2 is large, the conditional distribution of queue 1 approximates a Poisson distribution. The tail probability of queue 2 decays geometrically, while the tail probability of queue 1 decays much faster than queue 2’s. This helps us design an algorithm that computed the stationary distribution. Finally, by using the algorithm, we perform a sensitivity analysis on various system parameters, i.e., the arrival rates, service rates and the upgrade rate. The numerical study provides helpful insights into designing such service systems.

Communication-Aware Bandwidth-Optimized Predictive Control of Motor Drives in Electric Vehicles

To address the ever growing complexity of electric vehicles, several control functions can be integrated over a single shared intravehicle network, e.g., controller area network (CAN). However, sharing communication resources may lead to communication delays that affect control performance. Well-known communication standards allow to introduce different quality-of-service guarantees on a message-by-message basis. This study focuses on predictive control of motor drives; the model predictive control (MPC) approach is extended to jointly decide the value of each control command and its transmission priority on the CAN bus. A mixed integer quadratic problem is derived in which the transmission strategies are modeled by a two-class priority scheme and the corresponding values of delay variation are transformed into loss probabilities by introducing receiver buffers. System-level simulations show that the proposed approach uses high priority when the system behavior is farther from the desired one or bus is congested and the tracking error is reduced with respect to the traditional MPC and unprioritized transmission.

Rational Abandonment from Priority Queues: Equilibrium Strategy and Pricing Implications

Observable priority queues are prevalent in practice and create incentives for utility-maximizing customers to abandon after joining. However, the literature has so far ignored this behavior and the resulting system control issues. This paper studies the rational abandonment behavior of utility-maximizing customers in the context of an observable two-class priority queue, and identifies novel implications. We characterize the equilibrium abandonment strategy of low-priority customers and show that it has a threshold structure that depends on the fee structure. We then consider pricing as a means to control the balking and abandonment behavior, both under welfare maximization and revenue maximization. Our pricing results highlight the importance of the timing of payments. We show that welfare-maximization requires charging only a service fee and no entrance fee. In contrast, revenue maximization generally requires a combination of both an entrance and a service fee. This two-fee structure is equivalent to charging only upon entrance but offering a partial cancellation refund. Moreover, charging only an entrance fee may generate more or less revenue than charging only a service fee, but the performance of the latter policy is more robust. This appears to be the first paper that (i) gives an analytical characterization of equilibrium abandonment behavior in observable priority queues, and (ii) studies pricing for any queueing system in presence of rational customer abandonment.

Spatial–Temporal Queuing Theoretic Modeling of Opportunistic Multihop Wireless Networks With and Without Cooperation

In this paper, we characterize the average end-to-end delay in an opportunistic multihop secondary cognitive radio network overlaid with a primary multihop network. The nodes in both networks use a random medium access control scheme with exponentially distributed backoff. We first model the network as a two-class priority queuing network and use queuing theoretic approximation techniques to obtain a set of relations involving the mean and second moments of the interarrival and service times of packets at a secondary node. Then, applying these parameters to an equivalent open single-class G/G/1-queuing network, we obtain expressions for the average end-to-end delay of a packet in the secondary network using a diffusion approximation. Next, we extend the analysis to a case where secondary nodes cooperatively relay primary packets to improve their own transmission opportunities. The mathematical results are validated against extensive simulations.

Heavy-traffic asymptotics of a priority polling system with threshold service policy

In this paper, by the singular-perturbation technique, we investigate the heavy-traffic behavior of a priority polling system with three queues under threshold policy. It turns out that the scaled queue-length of the critically loaded queue is exponentially distributed, independent of that of the stable queues, which possess the same distributions as a two-class priority queue with N-policy vacation. Further, we provide an approximation of the tail queue-length distribution of the stable queues, which shows that it has the same prefactors and decay rates as the classical two-class preemptive priority queue. Stochastic simulations are taken to support the results.

On completion times in a two-class priority queue with impatience

In this note, we consider a two-class priority queueing system with Poisson arrivals, general service time distribution and one server, in which customers that are not currently being served may leave the queue according to exponentially distributed patience times, i.e., a M1, M2/G/1 + M system using a generalised Kendall’s notation. We expand the classic methodology to derive analytical formulas for the completion times in such a system, using preemptive repeat different and preemptive repeat identical disciplines. Known average completion times for priority queues without impatience are retrieved as limit cases.

Partially shared buffers with full or mixed priority

This paper studies a finite-sized discrete-time two-class priority queue. Packets of both classes arrive according to a two-class discrete batch Markovian arrival process (2-DBMAP), taking into account the correlated nature of arrivals in heterogeneous telecommunication networks. The model incorporates time and space priority to provide different types of service to each class. One of both classes receives absolute time priority in order to minimize its delay. Space priority is implemented by the partial buffer sharing acceptance policy and can be provided to the class receiving time priority or to the other class. This choice gives rise to two different queueing models and this paper analyses both these models in a unified manner. Furthermore, the buffer finiteness and the use of space priority raise some issues on the order of arrivals in a slot. This paper does not assume that all arrivals from one class enter the queue before those of the other class. Instead, a string representation for sequences of arriving packets and a probability measure on the set of such strings are introduced. This naturally gives rise to the notion of intra-slot space priority. Performance of these queueing systems is then determined using matrix-analytic techniques. The numerical examples explore the range of service differentiation covered by both models.

Performance analysis of a priority queue: Expedited Forwarding PHB in DiffServ

This paper studies a two-class priority queue in order to model a DiffServ router with Expedited Forwarding Per-Hop Behavior for high-priority traffic. Normally, queueing models assume infinite queue capacity but in a DiffServ router the capacity for high-priority traffic is often small to prevent this traffic from monopolizing the output link and hence causing starvation of other traffic. The presented model takes the exact (finite) high-priority queue capacity into account and introduces a framework for queue management algorithms, enabling determination of high-priority packet loss and its influence on the performance of the system. Analytical formulas for system content and packet delay of each traffic class are determined. Numerical examples indicate the necessity of modelling the high-priority queue as finite in order to accurately capture performance in a DiffServ environment.

Calculation of output characteristics of a priority queue through a busy period analysis

Abstract In this paper, we analyze some output characteristics of a discrete-time two-class priority queue by means of probability generating functions. Therefore, we construct a Markov chain which – after analysis – provides a.o. the probability generating functions of the lengths of the busy periods of both classes. It is furthermore shown how performance measures, related to the output process, are calculated from these functions. The queueing model is kept fairly simple to explain the method of analysis of the busy periods and the output characteristics of priority queues as clearly as possible.

Priority Assignment Under Imperfect Information on Customer Type Identities

In many service systems, customers are not served in the order they arrive, but according to a priority scheme that ranks them with respect to their relative “importance.” However, it may not be an easy task to determine the importance level of customers, especially when decisions need to be made under limited information. A typical example is from health care: When triage nurses classify patients into different priority groups, they must promptly determine each patient's criticality levels with only partial information on their conditions. We consider such a service system where customers are from one of two possible types. The service time and waiting cost for a customer depends on the customer's type. Customers' type identities are not directly available to the service provider; however, each customer provides a signal, which is an imperfect indicator of the customer's identity. The service provider uses these signals to determine priority levels for the customers with the objective of minimizing the long-run average waiting cost. In most of the paper, each customer's signal equals the probability that the customer belongs to the type that should have a higher priority and customers incur waiting costs that are linear in time. We first show that increasing the number of priority classes decreases costs, and the policy that gives the highest priority to the customer with the highest signal outperforms any finite class priority policy. We then focus on two-class priority policies and investigate how the optimal policy changes with the system load. We also investigate the properties of “good” signals and find that signals that are larger in convex ordering are more preferable. In a simulation study, we find that when the waiting cost functions are nondecreasing, quadratic, and convex, the policy that assigns the highest priority to the customer with the highest signal performs poorly while the two-class priority policy and an extension of the generalized cμ rule perform well.

On a generic class of two-node queueing systems

This paper analyzes a generic class of two-node queueing systems. A first queue is fed by an on---off Markov fluid source; the input of a second queue is a function of the state of the Markov fluid source as well, but now also of the first queue being empty or not. This model covers the classical two-node tandem queue and the two-class priority queue as special cases. Relying predominantly on probabilistic argumentation, the steady-state buffer content of both queues is determined (in terms of its Laplace transform). Interpreting the buffer content of the second queue in terms of busy periods of the first queue, the (exact) tail asymptotics of the distribution of the second queue are found. Two regimes can be distinguished: a first in which the state of the first queue (that is, being empty or not) hardly plays a role, and a second in which it explicitly does. This dichotomy can be understood by using large-deviations heuristics.

Analysis of M/M/1 queueing system with two priority classes

The present paper investigates the two-class priority M/M/1 queueing system, where the prioritized first class customers are served under first-come , first served (FCFS) preemptive resume discipline and have finite room capacity. The second class customers have infinite waiting space and low priority i.e. during the service of a low priority customer, if a high priority customer joins the system, then the low priority customer’s service is interrupted and will be resumed again when there is no high priority customer present in the system. Our aim is to get explicit expressions for the generating functions of the steady state probabilities in terms of Chebyshev’s polynomial of second kind have obtained. A recursive method is employed to solve the steady-state equations governing the model. Moreover, the expressions for the mean queue length, the marginal distributions of high and low priority customers are also given. The derivation is based only on the general structure of the system and the generating function involved, and thus is simpler than previous methods. We examine the effect of number of high priority customers, input and output rates on the average queue lengths of low and high priority classes by looking at a numerical illustration.

Two-class priority queueing system with restricted number of priority customers

In this paper, we analyze a two-class single-server preemptive priority queueing system with arrivals to each class are assumed to follow a Poisson process with exponentially distributed service times. Customers are served on a first-come, first-served basis within their own queue. Explicit expressions for the mean queue length and the steady-state joint distribution of the number of high and low priority customers in the system are derived. The analysis is based on the generating function technique. The obtained expressions are free from Bessel functions or any integral functions. Moreover, numerical values testing the quality of our analytical results are also presented.

Packet Models Revisited: Tandem and Priority Systems

We examine two extensions of traditional single-node packet-scale queueing models: tandem networks and (strict) priority systems. Two generic input processes are considered: periodic and Poisson arrivals. For the two-node tandem, an exact expression is derived for the joint distribution of the total queue length, and the queue length of the first queue, implicitly determining the distribution of the second queue. Similarly we derive the distribution of the low-priority queue in a two-class priority system. We also provide explicit approximations based on the Brownian bridge.

On the Two-Class M/M/1 System under Preemptive Resume and Impatience of the Prioritized Customers

The paper deals with the two-class priority M/M/1 system, where the prioritized class-1 customers are served under FCFS preemptive resume discipline and may become impatient during their waiting for service with generally distributed maximal waiting times. The class-2 customers have no impatience. The required mean service times may depend on the class of the customer. As the dynamics of class-1 customers are related to the well analyzed M/M/1+GI system, our aim is to derive characteristics for class-2 customers and for the whole system. The solution of the balance equations for the partial probability generating functions of the detailed system state process is given in terms of the weak solution of a family of boundary value problems for ordinary differential equations, where the latter can be solved explicitly only for particular distributions of the maximal waiting times. By means of this solution formulae for the joint occupancy distribution and for the sojourn and waiting times of class-2 customers are derived generalizing corresponding results recently obtained by Choi et al. in case of deterministic maximal waiting times. The latter case is dealt as an example in our paper.