Parallel Queues Research Articles

Towards the analysis of the performance measures of heterogeneous networks by means of two-phase queuing systems

Due to a multistage nature of transmission processes in heterogeneous 4G, 5G mobile networks, multiphase queuing systems become one of the most suitable ways for the resource allocation algorithms analysis and network investigation. In this paper, a few scientific papers that approached heterogeneous networks modelling by means of multiphase queuing systems are reviewed, mentioning the difficulties that arise with this type of analytical analysis. Moreover, several previously investigated models are introduced briefly as an example of two-phase systems of finite capacity and a special structure in discrete time that can be used for analysing resource allocation schemes based on the main performance measures obtained for wireless heterogeneous networks. One of the model presents a two-phase tandem queue with a group arrival flow of requests and a second phase of the complex structure that consists of parallel finite queues. The second model is a two-phase tandem queue with Markov modulated geometric arrival and service processes at the first phase and exhaustive service process at the second phase, which solves a cross-layer adaption problem in a heterogeneous network.

New Policy for Routing Jobs to Unobservable Parallel Queues with FIFO Scheduling

New Policy for Routing Jobs to Unobservable Parallel Queues with FIFO Scheduling

Waiting Time Before Justice in the Law Sector: A QUEUEING Theory

The legal system in Nigeria is be-deviled with delayed justice which has become a source of concern to many person. This is very much peculiar to those who feel that the judiciary is too slow in resolving legal issues in Nigeria. In Nigeria, there abound court cases especially those of criminal dimensions that have ongoing for years now without reaching a conclusive conclusion. The study is aimed at modelling the queuing system in the magistrate courts in Onitsha Magisterial districts. The specific objectives include: to apply the M/M/2 and M/M/3 models with identical and parallel queues to criminal cases in the Magistrate court and to compare the efficiency of the two models and in either cases to determine time to justice in criminal cases in the Magisterial District. The results of the study show that M/M/2 model with 2 identical and parallel queues have an expected number of cases in the system as 39 with 50% idle time while the M/M/3 model with 3 identical and parallel queues have an expected number of cases in the system as 23 cases with 64.88% idle time. The comparison of the two models shows M/M/2 with 2-identical and parallel queues is more efficient as it has more number of cases to attend to and less idle time. The study therefore concludes that the delays in the disposal of cases especially those with criminal nature may not be attributable to the queuing systems in place. Even though lesser number of servers is seen to be efficient, this may not be advised. Increasing the number of servers though will increase speed of disposal of cases, may also lead to increased idle time of servers. As more courts being set up may result in waste of resources, manpower and time as the 2-server system is efficient in speeding up justice delivery in the magisterial division.

Energy Efficient Task Scheduling in Fog Environment using Deep Reinforcement Learning Approach

The users of cloud span to several types of tasks for various purposes, such as users who need to accomplish tasks that utilize cloud based on as Infrastructure as a Service. These tasks are usually of high latency. There are some tasks that require immediate response. That is, ultra-low latency tasks like IoT device requirements. It is not feasible to always depend on a far cloud datacenter. Unlike traditional cloud computing, it is possible to place edge and fog nodes can be placed close to the IoT devices provides noticeable reduction in latency. The emerging fog computing technology is characterized by ultra-low latency response, which benefits several time-sensitive services and applications. Nodes in fog are deployed in less centralized. In a fog layer, the computing equipment dedicates parts of its limited resources to process IoT application tasks. Therefore, efficient utilization of computing resources is essential and requires an optimal and intelligent strategy for task scheduling. This paper focuses on scheduling IoT tasks in a fog-based environment with the aim to minimizing energy, cost, and service delay. Towards this end, a deep reinforcement learning based algorithm named Clipped Double Deep Q-learning using target networks and experience replay techniques is proposed. To ensure there is no lag in using resources optimally, a parallel queueing is utilized. One of the important factors in cloud (and fog) computing research is to address the problem of long waiting time of the task in the virtual machine queue. This paper proposes a dual queue method.

RESTART: ANALYSIS OF POST-COVID 19 CAPACITY IN SECURITY CHECKPOINTS AT MEXICO CITY AIRPORT

COVID-19 arrived in the world suddenly and unexpectedly. It caused major disruptions at economical, operational and other levels. In the case of flight traffic, the operations were reduced to 10% of their original levels. The question after COVID-19 is how to restart the operations and how to keep the balance between safety and capacity. In this paper we present an analysis using simulation techniques for understanding the impact in a security area of an important airport in Latin America; the airport of Mexico City. The results allow to illustrate the potential congestion given by the implemented covid-19 restriction, even when the traffic recovers only by 25% of the pre-covid-19 traffic. The congestion can be mitigated by applying some layout changes (snake queue vs parallel queue) and when more capacity is added to the system (extra security line). The results will raise situational awareness for airport stakeholders when implementing the actions suggested by different international institutions like WHO, IATA or ICAO.

Chaos-control and parallel queue synchronization of laser local area network

In this work, we study the chaos-control and parallel queue synchronization of a laser local area network (LAN). We present and study specifically a “single-queue-double-parameter” method of the parallel series queue dynamic behavior synchronization of the controlled laser LAN under two optoelectronic delay feedback controllers, and establish the mathematical and physical model of the controlled laser LAN. The LAN node is composed of two space coupled lasers with different parameters and other two single lasers, where two lasers series produce two different parallel queues, which results in two different chains of LAN nodes. Optical LAN links are composed of two optical parallel-crossing paths and two photoelectric delay feedback controllers setting in two lasers of LAN, which creates a method of double-parameter control of LAN. Through the analysis of the stability theory of differential equation and the dynamic characteristic equation of coupled lasers of LAN, our mathematical theory demonstrates that the chaos-control of laser LAN can be achieved by two photoelectric delay feedback controllers adjusting photoelectric feedback levels and feedback delay time of one of the two coupled laser and another single laser, respectively. Making analysis of the stability theory of differential equation and the dynamic characteristic equation of LAN nodes in two queue chains, we demonstrate theoretically how to obtain synchronization in network nodes of the controlled LAN on two queue chains by controlling optical feedback levels, and by the photoelectric delay feedback controllers adjusting photoelectric feedback levels and feedback delay time, respectively. Using our numerical calculation of parallel queue synchronization, the node laser’s waveform and its phase space trajectory, we find that very lasers of network nodes of the controlled LAN can lead to the parallel queue synchronization of a double-period, a three-period, a four-period and other quasi-periods while these quasi-periodic synchronizations and dynamic synchronizations are controlled in two queue chains of LAN nodes when we let the photoelectric feedback level and the delay time shift on some parameters. We find also two controlled quasi-periodic parallel queue synchronization regions. This paper also presents an application case of laser LAN multi-point chaotic carrier synchronous emission and ultra-wideband communication. This is a new type of controlled laser LAN system, which has the core elements of optical LAN and the characteristics of multi-variable, multi-dimension and parallel queue chaos-control techniques of complex dynamic networks. It also has the function of optical network ultra-wideband communication. The results have important reference value for studying the LAN, optical network and its synchronization and control, laser technology and chaos.

Synchronized Lévy queues

AbstractWe consider a multivariate Lévy process where the first coordinate is a Lévy process with no negative jumps which is not a subordinator and the others are non-decreasing. We determine the Laplace–Stieltjes transform of the steady-state buffer content vector of an associated system of parallel queues. The special structure of this transform allows us to rewrite it as a product of joint Laplace–Stieltjes transforms. We are thus able to interpret the buffer content vector as a sum of independent random vectors.

Analysis of Steady State Behaviour of Biserial and Parallel Queue Network Model with Batch Arrival

The primary objective of the paper is to analyse a network queue model with two queuing subsystems commonly allied to a single server. The first queue subsystem consists of two biserial servers while the other subsystem consists of three non serial parallel servers. At biserial queueing subsystem, the customers arrive in batches of fixed size from outside the system and individual arrival takes place at parallel subsystem. The input process is poisson and the service time distribution is exponential. Time independent solution and various queue characteristics have been calculated using generating function technique and laws of calculus. Numerical illustration is provided to have clear understanding of the model.

Optimal Control of Parallel Queues for Managing Volunteer Convergence

Volunteer convergence refers to the influx of volunteers to affected areas after large‐scale disasters. There are not only many benefits to volunteer convergence, but it also creates significant logistical challenges that can impede relief efforts. This study examines polices for admitting volunteers into organized relief operations, and for assigning admitted volunteers to relief tasks. We represent this problem as a queueing system where, in addition to customer arrivals and departures, random server arrivals and abandonments are also present. Then, using a Markov decision process framework, we analyze server admission and assignment policies that seek to minimize relief tasks holding costs as well as volunteer holding and rejection costs. We show that the classic c μ rule, a server allocation policy that determines where to put servers based on relief tasks holding costs and processing requirements, is optimal under both collaborative and non‐collaborative service regimes and when batch server arrivals are allowed. Additionally, we find that the optimal server admission policy is a complex state‐dependent policy. As a result, we propose a class of admission heuristics that depend on the number of workers in the system and the remaining system workload. In a numerical study, we show that our heuristic policies perform well with respect to long‐run average costs, waiting times, number of volunteers in the system, and number of volunteers idling in the system over a range of parameter values and distributions that are based on real data from a case study. As such, they promise volunteer coordinators an effective and simple way to manage disaster volunteers.

Stationary Waiting Time in Parallel Queues with Synchronization

Motivated by database locking problems in today’s massive computing systems, we analyze a queueing network with many servers in parallel (files) to which jobs (writing access requests) arrive according to a Poisson process. Each job requests simultaneous access to a random number of files in the database and will lock them for a random period of time. Alternatively, one can think of a queueing system where jobs are split into several fragments that are then randomly routed to specific servers in the network to be served in a synchronized fashion. We assume that the system operates on a first-come, first-served basis. The synchronization and service discipline create blocking and idleness among the servers, which leads to a strict stability condition compared with other distributed queueing models. We analyze the stationary waiting time distribution of jobs under a many-server limit and provide exact tail asymptotics. These asymptotics generalize the celebrated Cramér–Lundberg approximation for the single-server queue.

Information and pricing effects in two-tier public service systems

Public services such as healthcare may be offered by both free and toll service providers. Such a system, called a two-tier service system, is usually subject to congestion and often modeled as two parallel queues. Customers can choose one of three options: joining the free system, joining the toll system, or balking based on the congestion level in each service system and the toll price. This study examines the joint effects of delay information and the price of the toll system on system performance. We consider the performance measures from three different perspectives: the customers, toll service providers, and social planners. Two information scenarios are studied. In the first scenario, called a mixed information case, the free system does not offer real-time queue length information, contrary to the toll system. In the second scenario, called a non-real-time information case, neither system offers real-time queue length information. Because the system dynamics are highly complex, we propose a computational approach to analyze system performance based on Markovian queueing models. Numerical illustrations are presented to generate important managerial insights for decision-makers An important finding is that there are interaction effects of disclosing delay information and pricing toll service on customer equilibrium strategies and major performance measures of the system. To reduce the expected waiting time, the queue length information should be hidden for lower toll price range and should be disclosed for higher price range. It is also found that the revenue maximization price is different from the socially optimal price.

Heavy Traffic Limits for Join-the-Shortest-Estimated-Queue Policy Using Delayed Information

We consider a load-balancing problem for a network of parallel queues in which information on the state of the queues is subject to a delay. In this setting, adopting a routing policy that performs well when applied to the current state of the queues can perform quite poorly when applied to the delayed state of the queues. Viewing this as a problem of control under partial observations, we propose using an estimate of the current queue lengths as the input to the join-the-shortest-queue policy. For a general class of estimation schemes, under heavy traffic conditions, we prove convergence of the diffusion-scaled process to a solution of a so-called diffusion model, in which an important step toward this goal establishes that the estimated queue lengths undergo state-space collapse. In some cases, our diffusion model is given by a novel stochastic delay equation with reflection, in which the Skorokhod boundary term appears with delay. We illustrate our results with examples of natural estimation schemes, discuss their implementability, and compare their relative performance using simulations.

Parallel Queues with Discrete-Choice Arrival Pattern: Empirical Evidence and Asymptotic Characterization

We consider a parallel-queue system in which each queue is served by a dedicated service provider. The arrival process is driven by a discrete choice model, that is, customers observe the queue length for each service provider and choose one to join upon arrival. We assume that a customer's utility is difference between the service reward and the waiting cost, both of which are heterogeneous. Empirical analysis of the vehicle queues at the U.S.-Canada border-crossing port of entry supports our model setting. We show that with such a choice model, the arrival rate function satisfies certain properties, which allow us to characterizes the fluid and diffusion limit of the queue-length process. In particular, we show that even without the well-used Lipschitz-continuity assumption, the fluid limit process is unique and is attracted to a unique equilibrium. The diffusion limit process is a reflected multi-dimensional Ornstein-Uhlenbeck process centered at that equilibrium. We prove that the stationary distribution of the diffusion limit is a truncated multivariant Gaussian and interchange of limits holds.

Endogenous queue number determination in G/M/s systems

This paper presents a model for the endogenous determination of the number of queues in a G/M/s system. Customers arriving at a system where s customers are being served play a game, choosing between s parallel queues or one single queue. Equilibria are obtained for risk-neutral and risk-averse customers. With risk-neutral customers, both a single queue and multiple queues are equilibrium states. When risk-averse customers are considered, there is a unique single queue equilibrium. These results are discussed and suggestions for further research put forth.

Two‐class constrained optimization with applications to queueing control

AbstractConstrained Markov decision process (CMDP) is a methodology that has not seen wide applications in the literature, but is a more natural specification for modeling preferences in modern service systems. In this paper we present a general framework for solving two‐class CMDPs. In particular, we show that CMDPs can be solved by using the Lagrangian dual to specify a particular unconstrained problem. If an appropriate Lagrange multiplier can be discerned, structural results can be exploited to solve the original CMDP with the appropriate structure. We show that for two queues in parallel or two queues in series, the framework leads to simple threshold‐like optimal policies. The results in each case are used to develop heuristics for analogous problems with abandonments with applications to health care, call centers, and manufacturing systems. The efficacy of the heuristics is verified in each case via a detailed numerical study.

Multi Node Tandem Queuing Model with Binomial Bulk Size Distribution Having Load Dependent Service

In this article we study a multi node tandem queuing model consisting of K-nodes in which the customers arriving in batches to the first queue and after receiving service they will be directed with some node specific probability to join any one of the (K-1) parallel queues which are connected to first queue in series and exit from the system after getting service. It is assumed that the arrival and service completions follow Poisson processes and service rates depend on number of customers in the queue connected to it. Here the bulk arrivals are assumed to be Binomially distributed. Using difference differential equations the joint probability function is derived and performance measures such as average number of customers, waiting time of customer, throughput of each service station, utilization of each server, variance of number of customers in each queue are derived explicitly. A numerical illustration is provided to understand the theoretical results. Sensitivity analysis of the system behavior with regards to the arrival rates and load dependent service distribution parameters is carried out. A comparison between transient and study state behavior is also done .

A Hybrid Multiple Parallel Queuing Model to Enhance QoS in Cloud Computing

Among the different QoS metrics and parameters considered in cloud computing are the waiting time of cloud tasks, execution time of tasks in VM's, and the utilization rate of servers. The proposed model was developed to overcome some of the pitfalls in the existing systems among which are sub-optimal markdown in the queue length, waiting time, response time, and server utilization rate. The proposed model contemplates on the enhancement of these metrics using a Hybrid Multiple Parallel Queuing approach with a joint implementation of M/M/1: ∞ and M/M/s: N/FCFS to achieve the desired objectives. A neoteric set of mathematical equations have been formulated to validate the efficiency and performance of the hybrid queuing model. The results have been validated with reference to the workload traces of Bit Brains infrastructure provider. The results obtained indicate the significant reduction in the queue length by 60.93 percent, waiting time in the queue by 73.85 percent, and total response time by 97.51%.

Multi Node Tandem Queuing Model with Bulk Arrivals Having Geometric Arrival Distribution

In this paper a K-node forked queuing model with load dependent service rates is analysed. Here it is assumed that the customers arrive to the first queue in batches and wait for service. After getting service at first service station with some probability they may join any one of the (K-1) parallel queues which are connected to first queue in series and exit from the system after getting service. It is assumed that the arrival and service completions follow Poisson processes and service rates depend on number of customers in the queue connected to it. The influence of Geometrically distributed bulk arrivals on this queuing model is studied. Sensitivity analysis of the system behaviour with regards to the arrival rates and load dependent service distribution parameters is carried out. The influence of these parameters on system performance measures such as average number of customers, waiting time of customer, variation of number of customers in each queue, throughput of each service station, utilization of each server are derived explicitly when arrivals follow a Geometric distribution. Simulations are carried out to illustrate the result.

Martingales and buffer overflow for the symmetric shortest queue model

A variant of the standard symmetric system of two parallel queues under the join-the-shortest-queue policy is introduced. Here, the shortest queue has service rate $$\mu _1$$ , while the longest queue has rate $$\mu _2$$ , where $$\mu _1 + \mu _2 = 1$$ . In the case of equality, both queues are served at rate 1 / 2. Each queue has capacity K, which may be finite or infinite, and the global Poisson arrival rate is $$\rho $$ . Couplings show that, as $$\mu _2$$ varies, the model is totally ordered, in terms of both total number N(t) of customers in the system and longest queue length. The two extreme cases $$\mu _2= 0$$ and $$\mu _2= 1$$ then provide simple stochastic bounds for N(t) for arbitrary $$\mu _2$$ . The ordering partially extends to the enlarged model in which, whenever the shortest queue is empty, the idle server at that queue helps—or on the contrary disturbs—the other server. The previous bounds remain valid. In the extended setup, different martingales are next built from the infinite capacity process. Those lead to simple explicit formulations of the means and Laplace transforms of the hitting time of saturation, for the process with finite K started from any initial state. Asymptotics are then derived, as K gets large. Using one particular mean time, the stationary blocking probability is explicitly obtained, extending the result by Dester et al. regarding the standard symmetric model. Finally, the joint distribution of the time and state of the system at the first queue-length equality is expressed as a function of the inverse of a simple explicit matrix.

Open Problem—Load Balancing Using Delayed Information

Load-balancing algorithms are important for efficiently routing jobs in systems of parallel queues; however, there has been relatively little attention devoted to developing algorithms in the presence of messaging and/or routing delays. Given a system of parallel queues with infinite capacity buffers; first-come, first-serve service discipline; and a single stream of incoming jobs that are routed by a dispatcher upon arrival, it has been shown that join the shortest queue (JSQ) satisfies certain optimality properties, including minimizing mean wait time when the job sizes are exponentially distributed ( Winston 1977 , Ephremides et al. 1980 ) and state space collapse of the queue lengths under heavy traffic scaling for general service distributions ( Reiman 1984 ). However, implementation of JSQ uses up-to-date information about the state of the buffers, which requires instantaneous exchange of multiple messages between the dispatcher and the queues. This challenge has led to the development of efficient algorithms that require fewer messages, including join the shortest of d queues (JSQ(d); Vvedenskaya et al. 1996 , Mitzenmacher 2001 ), join the idle queue (JIQ; Badonnel and Burgess 2008 , Lu et al. 2011 ), and persistent idle load distribution (PILD; Atar et al. 2019a , b ); see van der Boor et al. (2018) for a survey of some recent results in the many-server limit. Although these algorithms do not use full information about the system, they still use up-to-date information about the state of some of the queues. Because of the physical separation between the dispatcher and the queues, processing effects, or periodic updates from the queues, the dispatcher may have access only to information about the delayed states of the queues. (There may also be routing delays because of the time it takes a job to travel between the dispatcher and its assigned server, for which these algorithms also do not account.) In such settings, it has been shown that JSQ can perform quite poorly and lead to sustained oscillations in queue lengths ( Mitzenmacher 2000 ).