Queue Length Distribution Research Articles

Dual core processors: Coupled queues: Transient performance evaluation

High-Performance Computing (HiPC) systems routinely employ multi/many – core processors. Specifically, dual – core processors find many applications in pervasive computing devices. Dual–core processors employ buffers for queueing the incoming jobs. Traditionally, the queues at the processors are assumed to be independent and the queueing system is analyzed in equilibrium for tractability purposes. Queues are modeled using Continuous Time Markov Chains (CTMC's) and the equilibrium performance measures are determined to analyze as well as design the queueing systems. In most interesting cases, the incoming jobs are routed to the queues using the Join the Shortest Queue (JSQ) policy. Thus, with such an adaptive routing algorithm, the two queues are evidently coupled and are not statistically independent. Hence traditional equilibrium performance evaluation doesn't provide realistic performance measures. In this research paper, the two queues associated with buffers in dual-core processors are considered to be coupled. The Coupled Queues are modeled using a Quasi – Birth – and – Death (QBD) process. Using traditional results related to QBD processes, equilibrium performance measures are determined. More interestingly, we demonstrate the tractability of the computation of transient probability distribution of a QBD process. In the research literature, transient analysis of the QBD process was shown to be tractable in the Laplace transform domain. But in this research paper, we prove that the matrix exponential eQt arising in transient analysis (where Q is the generator matrix of the QBD process) can be computed directly in the time domain rendering efficient transient analysis of QBD process. Using the transient probability mass function of queue length, estimation of transient performance measures such as expected queue length, average delay, and tail distribution can be determined. Further, optimal adaptive routing algorithms for coupled queues can be designed.

Performance Study of an M/M/1 Retrial Queueing System with Balking, Dissatisfied Customers, and Server Vacations

This study considers a single server retrial queueing system with dissatisfied customers, server vacations, and balking customers. When new arrivals clients see an idle server available, they immediately enter service. If a new arrival sees the server busy, that arrival may balk and leave the system. Otherwise, the new arrival will enter a retrial orbit and try again later. Customers who are dissatisfied with the service will be served again until the service is completed successfully. The server begins a vacation when the current customer completes service. Birth and death state transitions are given. Steady-state distributions for queue length are found. Furthermore, a recursive technique is used to evaluate probabilities. Several performance measures of the system are identified and effects of changes to parameters are observed.

Uncertainty Estimation of Connected Vehicle Penetration Rate

Knowledge of the connected vehicle (CV) penetration rate is crucial for realizing numerous beneficial applications during the prolonged transition period to full CV deployment. A recent study described a novel single-source data penetration rate estimator (SSDPRE) for estimating the CV penetration rate solely from CV data. However, despite the unbiasedness of the SSDPRE, it is only a point estimator. Consequently, given the typically nonlinear nature of transportation systems, model estimations or system optimizations conducted with the SSDPRE without considering its variability can generate biased models or suboptimal solutions. Thus, this study proposes a probabilistic penetration rate model for estimating the variability of the results generated by the SSDPRE. An essential input for this model is the constrained queue length distribution, which is the distribution of the number of stopping vehicles in a signal cycle. An exact probabilistic dissipation time model and a simplified constant dissipation time model are developed for estimating this distribution. In addition, to improve the estimation accuracy in real-world situations, the braking and start-up motions of vehicles are considered by constructing a constant time loss model for use in calibrating the dissipation time models. VISSIM simulation demonstrates that the calibrated models accurately describe constrained queue length distributions and estimate the variability of the results generated by the SSDPRE. Furthermore, applications of the calibrated models to the next-generation simulation data set and a simple CV-based adaptive signal control scheme demonstrate the readiness of the models for use in real-world situations and the potential of the models to improve system optimizations. Funding: This work was supported by The University of Hong Kong [Francis S Y Bong Professorship in Engineering and Postgraduate Scholarship] and by the Council of the Hong Kong Special Administrative Region, China [Grants 17204919 and 17205822]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/trsc.2023.1209 .

Heavy Traffic Queue Length Distribution without Resource Pooling in an Input-Queued Switch

This paper studies the heavy-traffic joint distribution of queue lengths of an input-queued switch operating under the MaxWeight scheduling policy. Input-queued switch acts as a representative of SPNs that do not satisfy the so-called complete resource pooling (CRP) condition, and consequently exhibit a multidimensional state space collapse. Except in special cases, only mean queue lengths of such non-CRP systems have been obtained in the literature. In this paper, we develop the transform method to study the steady state distribution of non-CRP systems. The key challenge is in solving an implicit functional equation involving the Laplace transform of the heavy-traffic limiting distribution. We then consider the general n - n input-queued switch that has n2 queues. Under a conjecture on uniqueness of the solution of the functional equation, we obtain an exact joint distribution of the heavy-traffic limiting queue-lengths in terms of a nonlinear transformation of 2n iid exponentials.

A Novel Computational Procedure for the Waiting-Time Distribution (In the Queue) for Bulk-Service Finite-Buffer Queues with Poisson Input

In this paper, we discuss the waiting-time distribution for a finite-space, single-server queueing system, in which customers arrive singly following a Poisson process and the server operates under (a,b)-bulk service rule. The queueing system has a finite-buffer capacity ‘N’ excluding the batch in service. The service-time distribution of batches follows a general distribution, which is independent of the arrival process. We first develop an alternative approach of obtaining the probability distribution for the queue length at a post-departure epoch of a batch and, subsequently, the probability distribution for the queue length at a random epoch using an embedded Markov chain, Markov renewal theory and the semi-Markov process. The waiting-time distribution (in the queue) of a random customer is derived using the functional relation between the probability generating function (pgf) for the queue-length distribution and the Laplace–Stieltjes transform (LST) of the queueing-time distribution for a random customer. Using LSTs, we discuss the derivation of the probability density function of a random customer’s waiting time and its numerical implementations.

Explicit Results for the Distributions of Queue Lengths for a Non-Preemptive Two-Level Priority Queue

Explicit Results for the Distributions of Queue Lengths for a Non-Preemptive Two-Level Priority Queue

Analysis of Non-Steady Queue-Length Distribution in a Finite-Buffer Model with Group Arrivals and Power Saving Mechanism with Setups

In the manuscript, a probability distribution of the queue length is studied in a model with group Markov arrivals, arbitrarily distributed service times and finite waiting room. After the period of suspension of service due to lack of packets, each new busy period is preceded by a random setup time. Integral equations for time-dependent queue-length distribution are derived by identifying renewal moments in the operation of the system and by applying total probability law. The representation for the solution of the system is found in terms of Laplace transforms. Computational examples illustrating the impact of system parameters on the queue-length distribution are included.

Analyzing a single hyper-exponential working vacation queue from its governing difference equation

AbstractAs the queue becomes exhausted, different maintenance tasks can be performed according to the fatigue load and wear degree of the service equipment. At the same time, considering the customer's sensitivity to time delay, the service facility will not completely remain inactive during the maintenance period. To describe this objectively existing phenomenon arising in the waiting line system, we consider a hyper-exponential working vacation queue with a batch renewal arrival process. Through the calculation of the well-structured roots of the associated characteristic equation, the shift operator method in the theory of difference equations and the supplementary variable technique for stochastic modeling plays a central role in the queue-length distribution analysis. Comparison with other ways to analyze queueing models, the advantage of our approach is that we can avoid deriving the complex transition probability matrix of the queue-length process embedded at input points. The feasibility of this approach is verified by extensive numerical examples.

The fixed-cycle traffic-light queue with multiple lanes and temporary blockages

Traffic-light modelling is a complex task, because many factors have to be taken into account. In particular, capturing all traffic flows in one model can significantly complicate the model. Therefore, several realistic features are typically omitted from most models. We introduce a mechanism to include pedestrians and focus on situations where they may block vehicles that get a green light simultaneously. More specifically, we consider a generalisation of the Fixed-Cycle Traffic-Light (FCTL) queue. Our framework allows us to model situations where (part of the) vehicles are blocked, e.g. by pedestrians that block turning traffic and where several vehicles might depart simultaneously, e.g. in case of multiple lanes receiving a green light simultaneously. We rely on probability generating function and complex analysis techniques which are also used to study the regular FCTL queue. We study the effect of several parameters on performance measures such as the mean delay and queue-length distribution.

Cost Optimization using Genetic Algorithm in Customers Intolerance Markovian Model with Working Vacation and Multiple Working Breakdowns

In this paper we consider a single server queueing model with working vacation and multiple working breakdowns. When on vacation, the server works at a different pace. Disturbances occur in the server due to multiple breakdowns. In working breakdowns server works at a different rate. During the time of interruption caused by working vacation and working breakdowns, the main server can find many implementations in operation. Both the server's lifespan and the time it takes to repair it are considered to be exponentially dispersed. Also, we have considered balking and reneging behaviours of customers. The stationary queue length distribution is computed using a matrix-analytic technique. Using Genetic Algorithm (GA) we optimize the cost function. The predicted length of a busy period, the expected length of a working vacation period, the expected length of a working breakdown period, the mean waiting time, and the average delay are all established. We compute numerical results to verify the analytical point of view. The effect of individual parameters is investigated using sensitivity analysis.

Asymptotic optimality of speed-aware JSQ for heterogeneous service systems

The Join-the-Shortest-Queue (JSQ) load-balancing scheme is known to minimise the average delay of jobs in homogeneous systems consisting of identical servers. However, it performs poorly in heterogeneous systems where servers have different processing rates. Finding a delay optimal scheme remains an open problem for heterogeneous systems. In this paper, we consider a speed-aware version of the JSQ scheme for heterogeneous systems and show that it achieves delay optimality in the fluid limit. One of the key issues in establishing this optimality result for heterogeneous systems is to show that the sequence of steady-state distributions indexed by the system size is tight in an appropriately defined space. The usual technique for showing tightness by coupling with a suitably defined dominant system does not work for heterogeneous systems. To prove tightness, we devise a new technique that uses the drift of exponential Lyapunov functions. Using the non-negativity of the drift, we show that the stationary queue length distribution has an exponentially decaying tail — a fact we use to prove tightness. Another technical difficulty arises due to the complexity of the underlying state-space and the separation of two time-scales in the fluid limit. Due to these factors, the fluid-limit turns out to be a function of the invariant distribution of a multi-dimensional Markov chain which is hard to characterise. By using some properties of this invariant distribution and using the monotonicity of the system, we show that the fluid limit is has a unique and globally attractive fixed point.

Analytic and Computational Analysis of GI/Ma,b/c Queueing System

Bulk-service queueing systems have been widely applied in many areas in real life. While single-server queueing systems work in some cases, multi-servers can efficiently handle most complex applications. Bulk-service, multi-server queueing systems (compared to well-developed single-server queueing systems) are more complex and harder to deal with, especially when the inter-arrival time distributions are arbitrary. This paper deals with analytic and computational analyses of queue-length distributions for a complex bulk-service, multi-server queueing system GI/Ma,b/c, wherein inter-arrival times follow an arbitrary distribution, a is the quorum, and b is the capacity of each server; service times follow exponential distributions. The introduction of quorum a further increases the complexity of the model. In view of this, a two-dimensional Markov chain has to be involved. Currently, it appears that this system has not been addressed so far. An elegant analytic closed-form solution and an efficient algorithm to obtain the queue-length distributions at three different epochs, i.e., pre-arrival epoch (p.a.e.), random epoch (r.e.), and post-departure epoch (p.d.e.) are presented, when the servers are in busy and idle states, respectively.

The Geo/Ga,Y/1/N Queue Revisited

We not only present an alternative and simpler approach to find steady-state distributions of the number of jobs for the finite-space queueing model Geo/Ga,Y/1/N using roots of the inherent characteristic equation, but also correct errors in some published papers. The server has a random service capacity Y, and it processes the jobs only when the number of jobs in the system is at least ‘a’, a threshold value. The main advantage of this alternative process is that it gives a unified approach in dealing with both finite- and infinite-buffer systems. The queue-length distribution is obtained both at departure and random epochs. We derive the relation between the discrete-time Geo/Ga,Y/1/N queue and its continuous-time analogue. Finally, we deal with performance measures and numerical results.

Group service system with three queues and load balancing

Abstract A group service system for three queues is considered. At each time t = 1, 2, . . ., with some probability, a customer enters the system, selects randomly two queues, and goes to the shorter one. At each moment such that there is at least one customer in each queue, each queue performs instantly the service of one customer. By means of Lyapunov functions, a criterion for the ergodicity of the Markov chain corresponding to this queuing system is established. The limiting joint distribution of queue lengths is found, and the connection with the problem of balanced allocations of particles into cells is described. In the corresponding problem of balanced allocation of particles, the limiting distribution of the range is found, i. e. the difference between the maximal and minimal numbers of particles in cells.

Simple analytical solutions for the , , and related queues

AbstractIn this paper we revisit some classical queueing systems such as the M $^b$ /E $_k$ /1/m and E $_k$ /M $^b$ /1/m queues, for which fast numerical and recursive methods exist to study their main performance measures. We present simple explicit results for the loss probability and queue length distribution of these queueing systems as well as for some related queues such as the M $^b$ /D/1/m queue, the D/M $^b$ /1/m queue, and fluid versions thereof. In order to establish these results we first present a simple analytical solution for the invariant measure of the M/E $_k$ /1 queue that appears to be new.

A load balancing system in the many-server heavy-traffic asymptotics

We study the load balancing system operating under Join the Shortest Queue (JSQ) in the many-server heavy-traffic regime. If $N$ is the number of servers, we let the difference between the total service rate and the total arrival rate be $N^{1-\alpha}$ with $\alpha>0$. We show that for $\alpha>4$ the average queue length behaves similarly to the classical heavy-traffic regime. Specifically, we prove that the distribution of the average queue length multiplied by $N^{1-\alpha}$ converges to an exponential random variable. Moreover, we show a result analogous to state space collapse. We provide two proofs for our result: one using the one-sided Laplace transform, and one using Stein's method. We additionally obtain the rate of convergence in the Wasserstein's distance.

Finite-Buffer M/G/1 Queues with Time and Space Priorities

Many communication systems have finite buffers and service delay-sensitive and loss-sensitive types of traffic simultaneously. To meet the diverse QoS requirements of these heterogeneous types of traffic, it is desirable to offer delay-sensitive traffic time priority over loss-sensitive traffic, and loss-sensitive traffic space priority over delay-sensitive traffic. To analyze the performance of such systems, we study a finite-buffer M/G/1 priority queueing model where nonpreemptive time priority is given to delay-sensitive traffic and push-out space priority is given to loss-sensitive traffic. Compared to the previous study on finite-buffer M/M/1 priority queues with time and space priority, where service times are identical and exponentially distributed for both types of traffic, in our model we assume that service times are different and are generally distributed for different types of traffic. As a result, our model is more suitable for the performance analysis of communication systems accommodating multiple types of traffic with different service-time distributions. For the proposed queueing model, we derive the queue-length distributions, loss probabilities, and mean waiting times of both types of traffic, as well as the push-out probability of delay-sensitive traffic. With numerical examples, we also explore how the performance measures are affected by system parameters such as the buffer size, and the arrival rates and mean service times of both types of traffic for different service-time distributions.

The generalized join the shortest orbit queue system: stability, exact tail asymptotics and stationary approximations

We introduce the generalized join the shortest queue model with retrials and two infinite capacity orbit queues. Three independent Poisson streams of jobs, namely a smart, and two dedicated streams, flow into a single-server system, which can hold at most one job. Arriving jobs that find the server occupied are routed to the orbits as follows: Blocked jobs from the smart stream are routed to the shortest orbit queue, and in case of a tie, they choose an orbit randomly. Blocked jobs from the dedicated streams are routed directly to their orbits. Orbiting jobs retry to connect with the server at different retrial rates, i.e., heterogeneous orbit queues. Applications of such a system are found in the modeling of wireless cooperative networks. We are interested in the asymptotic behavior of the stationary distribution of this model, provided that the system is stable. More precisely, we investigate the conditions under which the tail asymptotic of the minimum orbit queue length is exactly geometric. Moreover, we apply a heuristic asymptotic approach to obtain approximations of the steady-state joint orbit queue-length distribution. Useful numerical examples are presented and shown that the results obtained through the asymptotic analysis and the heuristic approach are agreed.

Performance analysis of an M/G/1 queue with bi-level randomized (p, N1, N2)-policy

This paper proposes an M/G/1 queueing model with bi-level randomized (p, N1, N2)-policy. That is, after all of the customers in the system are served, the server is closed down immediately. If N1(≥ 1) customers are accumulated in the queue, the server is activated for service with probability p(0 ≤ p ≤ 1) or still left off with probability 1 − p. When the number of customers in the system becomes N2(≥ N1), the server begins serving the waiting customers until the system becomes empty again. Using the total probability decomposition technique and the Laplace transform, we study the transient queue length distribution and obtain the expressions of the Laplace transform of the transient queue-length distribution with respect to time t. Then, employing L’Hospital’s rule and some algebraic operations, the explicit recursive formulas of the steady-state queue-length distribution, which can be used to accurately evaluate the probabilities of queue length, are presented. Moreover, some other important queuing performance indices, such as the explicit expressions of its probability generating function of the steady-state queue-length distribution, the expected queue size and so on, are derived. Meanwhile, we investigate the system capacity optimization design by the steady-state queue-length distribution. Finally, an operating cost function is constructed, and by numerical calculation, we find the minimum of the long-run average cost rate and the optimal bi-level threshold policy (N*1,N*2) that satisfies the average waiting time constraints.

A Probabilistic Approach to Growth Networks

Single-class closed queueing networks, consisting of infinite-server and single-server queues with exponential service times and probabilistic routing, admit product-from solutions. Such solutions, although seemingly tractable, are difficult to characterize explicitly for practically relevant problems due to the exponential combinatorial complexity of its normalization constant (partition function). In “A Probabilistic Approach to Growth Networks,” Jelenković, Kondev, Mohapatra, and Momčilović develop a novel methodology, based on a probabilistic representation of product-form solutions and large-deviations concentration inequalities, which identifies distinct operating regimes and yields explicit expressions for the marginal distributions of queue lengths. From a methodological perspective, a fundamental feature of the proposed approach is that it provides exact results for order-one probabilities, even though the analysis involves large-deviations rate functions, which characterize only vanishing probabilities on a logarithmic scale.