Discriminatory Processor Sharing Research Articles

Efficient and accurate Lyapunov function-based truncation technique for multi-dimensional Markov chains with applications to discriminatory processor sharing and priority queues

Online service providers aim to satisfy the tail performance requirements of customers through Service-Level Objectives (SLOs). One approach to ensure tail performance requirements is to model the service as a Markov chain and obtain its steady-state probability distribution. However, obtaining the distribution can be challenging, if not impossible, for certain types of Markov chains, such as those with multi-dimensional or infinite state-space and state-dependent transitions. Examples include M/M/1 with Discriminatory Processor Sharing (DPS) and preemptive M/M/c with multiple priority classes and customer abandonment.To address this fundamental problem, we propose a Lyapunov function-based state-space truncation technique that leverages moments or bounds on moments of the state variables. This technique allows us to obtain tight truncation bounds while ensuring arbitrary probability mass guarantees for the truncated chain. We highlight the efficacy of our technique for multi-dimensional DPS and M/M/c priority queue with abandonment, demonstrating a substantial reduction in state space (up to 74%) compared to existing approaches. Additionally, we present three practical use cases that highlight the applicability of our truncation technique by analyzing the performance of the DPS system.

Processor Sharing Queues With Impatient Customers and State-Dependent Rates

We study queues with impatient customers and Processor Sharing (PS) discipline as well as other variants of PS discipline, namely, Discriminatory Processor Sharing (DPS) and Generalized Processor Sharing (GPS) disciplines, where customers have deadlines until the end of service (DES). Customers arrive according to a state-dependent Poisson process and have general impatience. Customers have exponential service times with state-dependent service rates. Analytical methods based on simple Markov chains are given for the performance analysis of such queues. The principal measures of performance are the steady-state probability of missing deadline and the steady-state probability of blocking. Similar results are obtained for related queues with Random Order Service (ROS) discipline where customers have deadlines until the beginning of service (DBS). In view of a lack of exact analytical results for First Come First Served (FCFS) queues with state-dependent rates, a highly accurate approximation method is also given for these latter queues. The efficacy and accuracy of the approach are illustrated by some numerical examples and simulation experiments.

Economic feasibility of virtual operators in 5G via network slicing

The provision of services by more than one operator over a common network infrastructure, as enabled by 5G network slicing, is analyzed. Two business models to be implemented by a network operator, who owns the network, and a virtual operator, who does not, are proposed. In one business model, named strategic, the network operator provides service to its user base and the virtual operator provides service to its user base and pays a per-subscriber fee to the network operator. In the other business model, named monopolistic, the network operator provides service to both user bases.The two proposals are analyzed by means of a model that captures both system and economic features. As regards the systems features, the slicing of the network is modeled by means of a Discriminatory Processor Sharing queue. As regards the economic features, the incentives are modeled by means of the user utilities and the operators’ revenues; and game theory is used to model the strategic interaction between the users’ subscription decision and the operators’ pricing decision. In both business models, it is shown that the network operator can be provided with the appropriate economic incentives so that it acquiesces in serving the virtual operator’s user base (monopolistic model) and in allowing the virtual operator to provide service over the network operator’s infrastructure (strategic model). From the point of view of the users, the strategic model results in a higher subscription rate than the monopolistic model.

Fluid approximation of closed queueing networks with discriminatory processor sharing

As a multi-class variant of the classical egalitarian processor-sharing (EPS) discipline, discriminatory processor sharing (DPS) provides a suitable paradigm to model systems where share exists to control the service access of heterogeneous jobs. Although DPS is a more fine-grained scheduling discipline than EPS, the behavior of closed queueing networks (QNs) with DPS remains poorly understood. We propose in this paper a fluid approach to transient and steady-state analysis of closed QNs comprising delay and DPS stations. Our reference model features an arbitrary routing topology, phase-type service time distributions and class switching. The proposed approach has been validated against simulation for both transient and steady-state analysis. Moreover, we introduce a refined method for approximating response time distributions at either station or system level through transient analysis, showing its improvement upon prior art in handling non-exponentially distributed service times.

Heavy-traffic limits for Discriminatory Processor Sharing models with joint batch arrivals

We study the performance of Discriminatory Processor Sharing (DPS) systems, with exponential service times and in which batches of customers of different types may arrive simultaneously according to a Poisson process. We show that the stationary joint queue-length distribution exhibits state-space collapse in heavy traffic: as the load ρ tends to 1, the scaled joint queue-length vector (1−ρ)Q converges in distribution to the product of a deterministic vector and an exponentially distributed random variable, with known parameters. The result provides new insights into the behavior of DPS systems. It shows how the queue-length distribution depends on the system parameters, and in particular, on the simultaneity of the arrivals. The result also suggests simple and fast approximations for the tail probabilities and the moments of the queue lengths in stable DPS systems, capturing the impact of the correlation structure in the arrival processes. Numerical experiments indicate that the approximations are accurate for medium and heavily loaded systems.

Tighter Lyapunov Truncation for Multi-Dimensional Continuous Time Markov Chains with Known Moments

Continuous Time Markov chains (CTMCs) are widely used to model and analyze networked systems. A common analysis approach is to solve the system of balance equations governing the state transitions of a CTMC to obtain its steadystate probability distribution, and use the state probabilities to derive or compute various performance measures. In many systems, the state space of the underlying CTMC is infinite and multi-dimensional with state-dependent transitions; exact analysis of such models is challenging. For example, the exact probability distribution of the number of jobs in the Discriminatory Processor Sharing (DPS) system, first proposed by Kleinrock in 1967 [4], is still an open challenge. Likewise, obtaining the exact state probabilities of quasi-birth-and-death (QBD) processes with leveldependent transitions is known to be challenging [1]; QBDs are infinite state space multi-dimensional Markov chains in which states are organized into levels and transitions are skip-free between the levels.

On the equivalence between multiclass processor sharing and random order scheduling policies

Consider a single server system serving a multiclass population. Some popular scheduling policies for such system are the discriminatory processor sharing (DPS), discriminatory random order service (DROS), generalized processor sharing (GPS) and weighted fair queueing (WFQ). In this paper, we propose two classes of policies, namely MPS (multiclass processor sharing) and MROS (multiclass random order service), that generalize the four policies mentioned above. For the special case when the multiclass population arrive according to Poisson processes and have independent and exponential service requirement with parameter ?, we show that the tail of the sojourn time distribution for a class i customer in a system with the MPS policy is a constant multiple of the tail of the waiting time distribution of a class i customer in a system with the MROS policy. This result implies that for a class i customer, the tail of the sojourn time distribution in a system with the DPS (GPS) scheduling policy is a constant multiple of the tail of the waiting time distribution in a system with the DROS (respectively WFQ) policy.

Sojourn Time Approximations for a Discriminatory Processor Sharing Queue

We study a multiclass time-sharing discipline with relative priorities known as discriminatory processor sharing (DPS), which provides a natural framework to model service differentiation in systems. The analysis of DPS is extremely challenging, and analytical results are scarce. We develop closed-form approximations for the mean conditional (on the service requirement) and unconditional sojourn times. The main benefits of the approximations lie in its simplicity, the fact that it applies for general service requirements with finite second moments, and that it provides insights into the dependency of the performance on the system parameters. We show that the approximation for the mean conditional and unconditional sojourn time of a customer is decreasing as its relative priority increases. We also show that the approximation is exact in various scenarios, and that it is uniformly bounded in the second moments of the service requirements. Finally, we numerically illustrate that the approximation for exponential, hyperexponential, and Pareto service requirements is accurate across a broad range of parameters.

Markov-modulated M/G/1-type queue in heavy traffic and its application to time-sharing disciplines

This paper deals with a single-server queue with modulated arrivals, service requirements and service capacity. In our first result, we derive the mean of the total workload assuming generally distributed service requirements and any service discipline which does not depend on the modulating environment. We then show that the workload is exponentially distributed under heavy-traffic scaling. In our second result, we focus on the discriminatory processor sharing (DPS) discipline. Assuming exponential, class-dependent service requirements, we show that the joint distribution of the queue lengths of different customer classes under DPS undergoes a state-space collapse when subject to heavy-traffic scaling. That is, the limiting distribution of the queue length vector is shown to be exponential, times a deterministic vector. The distribution of the scaled workload, as derived for general service disciplines, is a key quantity in the proof of the state-space collapse.

Counterexamples to conjectures on discriminatory processor sharing and generalized processor sharing systems

In this paper, we provide counterexamples to conjectures, made by Verloop, Ayesta, and Borst (2010) [6], on discriminatory processor sharing (DPS) and generalized processor sharing (GPS) systems.

Approximate queue length distribution of a discriminatory processor sharing queue with impatient customers

We consider a two-class processor sharing queueing system with impatient customers. The system operates under the discriminatory processor sharing (DPS) scheduling. The arrival process of each class customers is the Poisson process and the service requirement of a customer is exponentially distributed. The reneging rate of a customer is a constant. To analyze the performance of the system, we develop a time scale decomposition approach to approximate the joint queue-length distribution of each class customers. Via a numerical experiment, we show that the time scale decomposition approach gives a fairly good approximation of the queue-length distribution and the expected queue length.

Invariance of the stationary distribution of states of networks with multimode service, different types of requests, and discriminatory processor sharing

Open and closed queuing networks with multimode strategies and several types of requests are studied. Single-line nodes can operate in several modes. The time spent in each mode has an exponential distribution. The queuing discipline of the server is the discriminatory processor sharing (DPS). The number of requests is a random variable with an arbitrary distribution function. The invariance is established of the stationary distribution of the probabilities of the network states with respect to the functional form of the distributions of the work required to service the requests.

Contributors

Contributors

In This Issue

In This Issue

Discriminatory processor sharing queues with stationary ergodic service times and the performance of TCP in overload

In a recent paper, Bonald and Roberts (2001) [6] studied non-persistent TCP connections in transient overload conditions, under the assumption that all connections have the same round-trip times. In this paper our goal is to develop theoretical tools that will enable us to relax this assumption and obtain explicit expressions for the rate of growth of the number of connections at the system, the rate at which TCP connections leave the system, as well as the time needed for the completion of a connection. To that end, we model the system as a discriminatory processor sharing (DPS) system which we analyze under very mild assumptions on the probability distributions related to different classes of arrivals: we only assume that the arrival rates of connections exist, and that the amount of information transmitted during a connection of a given type forms a stationary ergodic sequence. We then proceed to obtain explicit expressions for the growth rate of the number of connections at the DPS system for several specific probability distributions. We check through simulations the applicability of our queueing results for modeling TCP connections sharing a bottleneck.

Heavy-traffic analysis of the M/PH/1 discriminatory processor sharing queue with phase-dependent weights

We analyze a generalization of the Discriminatory Processor Sharing (DPS)queue in a heavy-traffic setting. Customers present in the system are served simultaneously at rates controlled by a vector of weights. We assume phase-type distributed service requirements and allow that customers have different weights in various phases of their service. We establish a state-space collapse for the queue length vector in heavy traffic. The result shows that in the limit, the queue length vector is the product of an exponentially distributed random variable and a deterministic vector. This generalizes a previous result by [2] who considered a DPS queue with exponentially distributed service requirements. We finally discuss some implications for residual service requirements and monotonicity properties in the ordinary DPS model.

Monotonicity Properties for Multi-Class Queueing Systems

We study multi-dimensional stochastic processes that arise in queueing models used in the performance evaluation of wired and wireless networks. The evolution of the stochastic process is determined by the scheduling policy used in the associated queueing network. For general arrival and service processes, we give sufficient conditions in order to compare sample-path wise the workload and the number of users under different policies. This allows us to evaluate the performance of the system under various policies in terms of stability, the mean overall delay and the mean holding cost. We apply the general framework to linear networks, where users of one class require service from several shared resources simultaneously. For the important family of weighted ?-fair policies, stability results are derived and monotonicity of the mean holding cost with respect to the fairness parameter ? and the relative weights is established. In order to broaden the comparison results, we investigate a heavy-traffic regime and perform numerical experiments. In addition, we study a single-server queue with two user classes, and show that under Discriminatory Processor Sharing (DPS) or Generalized Processor Sharing (GPS) the mean overall sojourn time is monotone with respect to the ratio of the weights. Finally we extend the framework to obtain comparison results that cover the single-server queue with an arbitrary number of classes as well.

Heavy-Traffic Analysis of a Multiple-Phase Network with Discriminatory Processor Sharing

We analyze a generalization of the discriminatory processor-sharing (DPS) queue in a heavy-traffic setting. Customers present in the system are served simultaneously at rates controlled by a vector of weights. We assume that customers have phase-type distributed service requirements and allow that customers have different weights in various phases of their service. In our main result we establish a state-space collapse for the queue-length vector in heavy traffic. The result shows that in the limit, the queue-length vector is the product of an exponentially distributed random variable and a deterministic vector. This generalizes a previous result by Rege and Sengupta [Rege, K. M., B. Sengupta. 1996. Queue length distribution for the discriminatory processor-sharing queue. Oper. Res. 44(4) 653–657], who considered a DPS queue with exponentially distributed service requirements. Their analysis was based on obtaining all moments of the queue-length distributions by solving systems of linear equations. We undertake a more direct approach by showing that the probability-generating function satisfies a partial differential equation that allows a closed-form solution after passing to the heavy-traffic limit. Making use of the state-space collapse result, we derive interesting properties in heavy traffic: (i) For the DPS queue, we obtain that, conditioned on the number of customers in the system, the residual service requirements are asymptotically independent and distributed according to the forward recurrence times. (ii) We then investigate how the choice for the weights influences the asymptotic performance of the system. In particular, for the DPS queue we show that the scaled holding cost reduces as classes with a higher value for dk/E(Bkfwd) obtain a larger share of the capacity, where dk is the cost associated to class k, and E(Bkfwd) is the forward recurrence time of the class-k service requirement. The applicability of this result for a moderately loaded system is investigated by numerical experiments.

Mean sojourn times for phase-type discriminatory processor sharing systems

In a discriminatory processor sharing (DPS) queueing model, each job (or customer) belongs to one out of finitely many classes. The arrival processes are Poisson. Classes differ with respect to arrival rates and service time distributions. Moreover, classes have different priority levels. All jobs present are served simultaneously but the fraction of the server’s capacity allocated to each one of them is proportional to their class priority parameter (while the total capacity is of course fixed). For the case of exponential service requirements, we find the mean sojourn times of various job classes, conditioning on the system’s state, namely how many of each class are present. Next we derive the well-known system of linear equations solved by the unconditional values. We then switch to the much weaker assumption of phase-type service requirements, different across classes, where additionally the priority parameters may depend both on class and phase (and not only on class as assumed in existing papers). We determine mean sojourn times for a job belonging to any class, firstly conditional on the numbers of jobs from all classes present (which are shown to be affine functions) and secondly, unconditionally (as solutions of a system of linear equations).

Discriminatory Processor Sharing Queues and the DREB Method

We study a discriminatory processor-sharing (DPS) queue with a Markovian arrival process (MAP) and K classes of customers. This system can be modeled into a K-dimensional quasi-birth-and death (QBD) process by the standard matrix-analytical approach, but such a process is computationally difficult to handle. We present a different formulation by which the K-dimensional QBD process is reduced to a single-dimensional level-dependent QBD process with expanding blocks. We derive analytical expressions and use them to compute the sojourn time and joint queue length distributions efficiently. This method allows us to study numerically a wide range of important open queueing problems that are common in computer and communication systems, such as various priority queues, queueing networks, and other dependent queues.