Exponential Service Research Articles

Service Rate and Admission Control in an M/G/1 Queue with State-Dependent Arrival Rates

Admission and service rate control problems in queueing systems have been studied in the literature. For an exponential service time distribution, the optimality of the threshold-type policy has been proved. However, in production systems, the production time follows a general distribution, not an exponential one. In this paper, control of the service speed according to the number of customers is considered. The analytical results of an M/G/1 queue with arrival and service rates that depend on the number of customers in the system, which is called an Mn/Gn/1 queue, are used to compute the performance measure of service rate control. In particular, for the case in which the arrival rates are the same among queue-length intervals, a computation method for deriving stationary distributions is developed. Constant, uniform, exponential, and Bernoulli distributions on the service time are considered via numerical experiments. The results show that the optimal threshold depends on the type of distribution, even if the mean value of the service time is the same. In addition, when the reward rate is small, a case in which a non-threshold-type service rate control policy outperforms all threshold-type policies is identified.

Influence of various application types on the performance of LTE mobile networks

Modern mobile internet networks are becoming heavier and denser. Also it is not regularly planned, and becoming more heterogeneous. The explosive growth in the usage of smartphones poses numerous challenges for LTE cellular networks design and implementation. The performance of LTE networks with bursty and self-similar traffic has become a major challenge. Accurate modeling of the data generated by each connected wireless device is important for properly investigating the performance of LTE networks. This paper presents a mathematical model for LTE networks using queuing theory considering the influence of various application types. Using sporadic source traffic feeding to the queue of the evolved node B and with the exponential service time assumption, we construct a queuing model to estimate the performance of LTE networks. We use the performance model presented in this paper to study the influence of various application categories on the performance of LTE cellular networks. Also we validate our model with simulation using NS3 simulator with different scenarios.

Optimal control of a queue under a quality-of-service constraint with bounded and unbounded rates

We consider a simple Markovian queue with Poisson arrivals and exponential service times for jobs. The controller chooses state-dependent service rates from an action space. The queue has a finite buffer, and when full, new jobs get rejected. The controller’s objective is to choose optimal service rates that meet a quality-of-service constraint. We solve this problem analytically and compute it numerically under two cases: When the action space is unbounded and when it is bounded.

The Age of Information in Networks: Moments, Distributions, and Sampling

A source provides status updates to monitors through a network with state defined by a continuous-time finite Markov chain. An age of information (AoI) metric is used to characterize timeliness by the vector of ages tracked by the monitors. Based on a stochastic hybrid systems (SHS) approach, first order linear differential equations are derived for the temporal evolution of both the moments and the moment generating function (MGF) of the age vector components. It is shown that the existence of a non-negative fixed point for the first moment is sufficient to guarantee convergence of all higher order moments as well as a region of convergence for the stationary MGF vector of the age. The stationary MGF vector is then found for the age on a line network of preemptive memoryless servers. From this MGF, it is found that the age at a node is identical in distribution to the sum of independent exponential service times. This observation is then generalized to linear status sampling networks in which each node receives samples of the update process at each preceding node according to a renewal point process. For each node in the line, the age is shown to be identical in distribution to a sum of independent renewal process age random variables.

Quality control for a detected an appropriate queue from K‐independent M/M/C/N queueing models

AbstractIn this paper, we consider a system of K‐independent Markovian queues such that each one of them has a Poisson arrival process and exponential service time. We assume that every server has some characteristics such as the speed of the service performance or the service cost. To find an appropriate queue, which meets customer needs for the service performance, we present a new approach that gives a suitable decision to choose an appropriate queue from our system. This allows the customer to deal with minimum cost and faster server under steady state. We solve an interesting discrete stochastic optimization problem where the paid cost by the customer is bounded by a Gaussian distribution. Using these hypotheses, we perform a simulation study by generating the paid cost random values and choosing the minimum value between them. This minimum cost gives the highest service rate, which is used to obtain the optimum values of the system effectiveness measures.

Stochastic modeling of parallel process flows in intra-logistics systems: Applications in container terminals and compact storage systems

Many intra-logistics systems, such as automated container terminals, distribution warehouses, and cross-docks, observe parallel process flows, which involve simultaneous (parallel) operations of independent resources while processing a job. When independent resources work simultaneously to process a common job, the effective service requirement of the job is difficult to estimate. For modeling simplicity, researchers tend to assume sequential operations of the resources. In this paper, we propose an efficient modeling approach for parallel process flows using two-phase servers. We develop a closed queuing network model to estimate system performance measures. Existing solution methods can evaluate the performance of closed queuing networks that consist of two-phase servers with exponential service times only. To solve closed queuing networks with general two-phase servers, we propose new solution methods: an approximate mean value analysis and a network aggregation dis-aggregation approach. We derive insights on the accuracy of the solution methods from numerical experiments. Although both solution methods are quite accurate in estimating performance measures, the network aggregation dis-aggregation approach consistently performs best. We illustrate the proposed modeling approach for two intra-logistic systems: a container terminal with automated guided vehicles and a shuttle-based compact storage system. Results show that approximating the simultaneous operations as sequential operations underestimates the container terminal throughput on average by 28% and at maximum up to 47%. Similarly, considering sequential operations of the resources in the compact storage system results in an underestimation of the throughput capacity up to 9%.

The Limit of Stationary Distributions of Many-Server Queues in the Halfin–Whitt Regime

We consider the so-called GI/GI/N queue, in which a stream of jobs with independent and identically distributed service times arrive as a renewal process to a common queue that is served by [Formula: see text] identical parallel servers in a first-come, first-served manner. We introduce a new representation for the state of the system and, under suitable conditions on the service and interarrival distributions, establish convergence of the corresponding sequence of centered and scaled stationary distributions in the so-called Halfin–Whitt asymptotic regime. In particular, this resolves an open question posed by Halfin and Whitt in 1981. We also characterize the limit as the stationary distribution of an infinite-dimensional, two-component Markov process that is the unique solution to a certain stochastic partial differential equation. Previous results were essentially restricted to exponential service distributions or service distributions with finite support, for which the corresponding limit process admits a reduced finite-dimensional Markovian representation. We develop a different approach to deal with the general case when the Markovian representation of the limit is truly infinite dimensional. This approach is more broadly applicable to a larger class of networks.

Математическая модель процесса потребления вычислительных ресурсов при их неполном освобождении

The article develops a mathematical model of the process of consuming computing resources when they are incompletely released using virtualization technology. The mathematical model is presented in the form of a queuing system with an unlimited number of devices, with a simple incoming flow of applications and exponential service time for them on the devices. The study of the model is performed by the methods of queuing theory. Using the method of moments, the main probabilistic characteristics of the amount of free resources are found mathematical expectation and variance. The proposed mathematical model of the process of consuming computing resources allows us to evaluate and predict the process of changing the amount of free resources of a virtual machine in time and analyze its performance parameters.

Вывод уравнений типа Колмогорова - Чепмена с оператором Фоккера - Планка

In this paper we obtain the differential equation of the type Kolmogorov-Chapman with differential operator of the Fokker-Planck, having theoretical and practical value in the differential equations theory. Equations concerning non-stationary and stationary characteristics of the number of applications obtained for a class of Queuing systems (QS) with an infinite storage device, one service device with exponential service, the input of which is supplied twice stochastic a Poisson flow whose intensity is a random diffusion process with springy boundaries and a non-zero drift coefficient. Service systems with diffusion intensity of the input flow are used for modeling of global computer networks nodes.

A lower bound on the queueing delay in resource constrained load balancing

We consider the following distributed service model: jobs with unit mean, general distribution, and independent processing times arrive as a renewal process of rate $\lambda n$, with $0<\lambda <1$, and are immediately dispatched to one of several queues associated with $n$ identical servers with unit processing rate. We assume that the dispatching decisions are made by a central dispatcher endowed with a finite memory, and with the ability to exchange messages with the servers. We study the fundamental resource requirements (memory bits and message exchange rate), in order to drive the expected queueing delay in steady-state of a typical job to zero, as $n$ increases. We develop a novel approach to show that, within a certain broad class of “symmetric” policies, every dispatching policy with a message rate of the order of $n$, and with a memory of the order of $\log n$ bits, results in an expected queueing delay which is bounded away from zero, uniformly as $n\to \infty $. This complements existing results which show that, in the absence of such limitations on the memory or the message rate, there exist policies with vanishing queueing delay (at least with Poisson arrivals and exponential service times).

Dynamic Resource Allocation in Fork-Join Queues

&lt;?tight?&gt;Fork-join systems play a pivotal role in the analysis of distributed systems, telecommunication infrastructures, and storage systems. In this article, we consider a fork-join system consisting of K parallel servers, each of which works on one of the K tasks that form each job. The system allocates a fixed amount of computational resources among the K servers, hence determining their service speed. The goal of this article is that of studying the resource allocation policies among the servers. We assume that the queueing disciplines of the fork- and join-queues are First Come First Served. At each epoch, at most K tasks are in service while the others wait in the fork-queues. We propose an algorithm with a very simple implementation that allocates the computational resources in a way that aims at minimizing the join-queue lengths, and hence at reducing&lt;?brk?&gt; the expected job service time. We study its performance in saturation and under exponential service time. The model has an elegant closed-form stationary distribution. Moreover, we provide an algorithm to numerically or symbolically derive the marginal probabilities for the join-queue lengths. Therefore, the expressions for the expected join-queue length and the expected response time under immediate join can be derived. Finally, we compare the performance of the proposed resource allocation algorithm with that of other strategies.

Sequencing in an appointment system with deterministic arrivals and non-identical exponential service times

This paper investigates an appointment system with deterministic arrival times and non-identical exponential service times with the objective of minimizing the expected costs of customer-waiting and server-idle times (WIT). For two customers, this paper shows analytically that the smallest-variance-first-rule (SV) and, equivalently, the smallest-mean-first-rule (SM) minimize WIT when the second customer arrives at either optimal or arbitrary arrival time. For three customers, this paper shows analytically that either SV or SM minimize WIT, assuming that each customer arrives at the cumulative sum of expected service times of prior customers. Based on numerical evaluation, this paper recommends that the exponential distribution parameter, which determines either SV or SM sequences, be used for a general number of customers.

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.

On Applications of the Earliest Due Date Rule in Stochastic Scheduling and in Queueing

We consider an environment with m servers in parallel and n jobs. The n jobs have i.i.d. exponentially distributed processing requirements. The servers operate at different speeds and preemptions are allowed. The jobs have random release dates that are not known in advance, but whenever a job is released, its due date is fixed and becomes known to the scheduler. We first consider three stochastic scheduling problems with three due date related objective functions and consider variations of the Earliest Due Date (EDD) rule, including the preemptive policy that at any point in time assigns the job with the Earliest Due Date to the Fastest Server (EDD-FS). Our optimality results turn out to be examples of stochastic scheduling problems that have relatively simple priority rules that are optimal while their deterministic counterparts do not allow such priority rules to be optimal. We furthermore extend our results to include a priority queueing model with m exponential servers that operate at different speeds and preemptions being allowed. The jobs arrive according to an arbitrary renewal process, and we assume that the utilization factor of the system is less than 1. Upon a job’s arrival the time difference between its due date and arrival date is set by the draw of a random variable from a given distribution. At a job’s arrival date, its due date is immediately fixed and known while its actual processing time only becomes known upon its service completion. We show that for this priority queueing model the preemptive EDD-FS rule also minimizes the limit of several due date related objective functions as the number of jobs converges to infinity.

Towards a performance analysis of composite web services using Petri nets

The proliferation of web services-based applications, collaboration, interoperability between companies, and more generally demands of web services over the internet are major challenges in the design of performance evaluation in web services discovery and composition. It is required for the web services providers to analyse the performance related to the satisfaction of clients requests. This paper presents an analytical model based on Petri nets, for evaluating the performance of a web services system where requests and web services follow an exponential server. The arrival process of the users requests and web services follows a Poisson distribution. When the web services finish the service, they join the queue. This model follows a Poisson distribution, it is based on Petri nets. We use an analytical method for solving the model. Furthermore, we compute the system's response time, and the average number of clients in the system in terms of the arrival rate of clients requests. We find the limit number of clients in the system from which it begins to be saturated.

Towards a Performance Analysis of Composite Web Services using Petri Nets.

The proliferation of web services-based applications, collaboration, interoperability between companies, and more generally demands of web services over the internet are major challenges in the design of performance evaluation in web services discovery and composition. It is required for the web services providers to analyse the performance related to the satisfaction of clients requests. This paper presents an analytical model based on Petri nets, for evaluating the performance of a web services system where requests and web services follow an exponential server. The arrival process of the users requests and web services follows a Poisson distribution. When the web services finish the service, they join the queue. This model follows a Poisson distribution, it is based on Petri nets. We use an analytical method for solving the model. Furthermore, we compute the system's response time, and the average number of clients in the system in terms of the arrival rate of clients requests. We find the limit number of clients in the system from which it begins to be saturated.

Queues With the Dropping Function and Non-Poisson Arrivals

We deal with the single-server queueing system, in which an arriving job (packet, customer) is not allowed to the queue with the probability depending on the queue size. Such a rejected job is lost and never returns to the queue. The study is motivated, but not limited to, active queue management in Internet routers. The exponential service times and general interarrival times are assumed, what makes the model to be a generalization of classic G/M/1 and G/M/1/N queueing models. Firstly, a replacement for the $\rho stability condition, which is too excessive in the considered system, is proven. Then, several popular performance characteristics are derived, including the distribution of the queue size, waiting time, workload and the time to reach a given level, as well as the loss ratio. Finally, numerical examples are presented, demonstrating the impact of the standard deviation of the interarrival time on the system performance, as well as the performance of the system for different parameterizations of the dropping function.

Optimal Pricing Analysis of Computer Networks Based on a Queueing System With Retrial Mechanism

We consider a single-server retrial queue with a Poisson arrival process and exponential service times, where the server is unreliable. Assume there is no waiting space in front of the server and the customer who finds the server unavailable joins an orbit to access the server some time later. We discuss two types of customers’ retrial behavior. One is that each customer in the orbit seeks for services independently and the total retrial rate of the system depends on the number of customers in the orbit. The other type of retrial discipline is called constant retrial policy and it arises from some situations in the computer and communication network where the retrial rate may be controlled by automatic mechanisms. An announced price charged by the server is imposed on customers joining the system and the actual demands for services depend on the price via a decreasing function. We investigate the system characteristics and study how the manager, whose goal is to maximize its own profit, determines the price charging joining customers. Finally, we present an application example to illustrate the obtained results and make comparisons between the two retrial policies from the perspective of customers’ expected waiting time.

A Model for Overflow Queuing Network with Two Stations Heterogeneous System

Network systems with overflow arrivals are the utmost imperative problems in high priority queuing system, where the customers are reluctant to join the queue. If the servers are free then arrived customers are serviced by one of the multiple servers. If all servers are busy then arrived customers will not join in the queue and quit the system without service and never return for the service. If the main server completes its work and no customer arrives for service then the work assigned to the lower service server is transformed to the main server and it becomes free for next arrival customer. In this paper, we studied two heterogeneous nodes queuing system with arrivals as a Poisson process and the service with exponential service times. The system is analysed using spectral expansion method to study various parametric measures and the results are presented in the graphical form.

A NEW LOOK ON THE SHORTEST QUEUE SYSTEM WITH JOCKEYING

AbstractWe introduce a Markov queueing system with Poisson arrivals, exponential services, and jockeying between two parallel and equivalent servers. An arriving customer admits to the shortest line. Every transition, of only the last customer in line, from the longer line to the shorter line may accompanied by a certain fixed cost. Thus, a transition from the longer queue to the shorter queue occurs whenever the difference between the lines reaches a certain discrete threshold (d = 2, 3, …). In this study, we focus on the stochastic analysis of the number of transitions of an arbitrary customer.