Exponential Service Times Research Articles

Queues with Redundancy

A major advantage of cloud computing and storage is the large-scale sharing of resources, which provides scalability and flexibility. But resource-sharing causes variability in the latency experienced by the user, due to several factors such as virtualization, server outages, network congestion etc . This problem is further aggravated when a job consists of several parallel tasks, because the task run on the slowest machine becomes the latency bottleneck. A promising method to reduce latency is to assign a task to multiple machines and wait for the earliest to finish. Similarly, in cloud storage systems requests to download the content can be assigned to multiple replicas, such that it is sufficient to download any one replica. Although studied actively in systems in the past few years, there is little work on rigorous analysis of how redundancy affects latency. The effect of redundancy in queueing systems was first analyzed only recently in [2, 3, 6], assuming exponential service time. General service time distribution, in particular the effect of its tail, is considered in [7, 8]. This work analyzes the trade-off between latency and the cost of computing resources in queues with redundancy, without assuming exponential service time. We study a generalized fork-join queueing model where finishing any k out of n tasks is sufficient to complete a job. The redundant tasks can be canceled when any k tasks finish, or earlier, when any k tasks start service. For the k = 1 case, we get an elegant latency and cost analysis by identifying equivalences between systems without and with early redundancy cancellation to M/G/1 and M/G/n queues respectively. For general k, we derive bounds on the latency and cost. Please see [4] for an extended version of this work.

Transient analysis of the Erlang A model.

We consider the Erlang A model, or M/M/m+M queue, with Poisson arrivals, exponential service times, and m parallel servers, and the property that waiting customers abandon the queue after an exponential time. The queue length process is in this case a birth–death process, for which we obtain explicit expressions for the Laplace transforms of the time-dependent distribution and the first passage time. These two transient characteristics were generally presumed to be intractable. Solving for the Laplace transforms involves using Green’s functions and contour integrals related to hypergeometric functions. Our results are specialized to the M/M/infty queue, the M / M / m queue, and the M / M / m / m loss model. We also obtain some corresponding results for diffusion approximations to these models.

A closed-form solution for a tollbooth tandem queue with two heterogeneous servers and exponential service times

This paper considers a tollbooth queueing system where customers arrive according to a Poisson process and there are two heterogeneous servers of exponential service times. We show that eigenvalues can be found explicitly for the characteristic matrix polynomial associated with the Markov chain characterizing the system. We derive a closed-form solution for the steady state probabilities to make the straightforward computation of performance measures.

Transient analysis of an M/M/1 queue with impatient behavior and multiple vacations

In this paper, we carry out an analysis for a single server queue with impatient customers and multiple vacations where customers impatience is due to an absentee of servers upon arrival. Customers arrive at the system according to a Poisson process and exponential service times. Explicit expressions are obtained for the time dependent probabilities, mean and variance of the system size in terms of the modified Bessel functions, by employing the generating functions along with continued fractions and the properties of the confluent hypergeometric function. Finally, some numerical illustrations are provided.

Analysis of approximately balanced production lines

This paper develops two analytical formulas for estimating the throughput of a reliable production line with exponential service times and finite intermediate buffers. The formulas apply in the case of an approximately balanced line with identical buffers or near optimal buffer allocations, where the processing times of the machines are close to each other but not necessarily the same. The derivation of the formulas is based on the standard decomposition method. Moreover, it is proved that, in general cases, both formulas provide upper bounds for the throughput obtained by the decomposition method. Numerical experiments show that the proposed formulas achieve good accuracy for approximately balanced production lines. Finally, the formulas are applied to the buffer allocation problem, and two closed-form expressions are obtained for estimating the smallest buffer capacity which is necessary to achieve the desired throughput.

Networks of queues models with several classes of customers and exponential service times

The main target of this paper is to present the Markov chain C that, not giving explicitly the queue lengths stationary probabilities, has the necessary information to its determination for open networks of queues with several classes of customers and exponential service times, allowing to overcome ingeniously this problem. The situation for closed networks, in the same conditions, much easier is also presented.

A Study on a Tandem Stochastic Queueing Model with Parallel Phases and a Numerical Example

In this study a two stage queueing model is analyzed. At first stage there is a single server having exponential service time with parameter μ_1 and no waiting is allowed in front of this server. There are two parallel phase-type servers at second stage and these parallel servers have exponential service time with parameter μ_2. Arrivals to this system is Poisson with parameter λ. An arriving customer to this system has service if the server at first stage is available or leaves the system if the server is busy where the first loss occurs. After having service in first stage the customer proceeds to the second stage, if both of the phase-type parallel servers in second stage are available the customer chooses one of these servers with probability 0.50 or leaves the system if any of these servers in second stage is busy so the second loss occurs. A customer who has service at both stages leaves the system. The number of customers in this model is represented by a 3-diamensional Markov chain and Kolmogorov differential equations are obtained. After that mean number of customers and mean waiting time in the system is obtained by limit probabilities. We have shown that the customer numbers at first and second stages are dependent to each other. The numerical analysis of obtained performance measures are shown by a numeric example. Finally the graphs of loss probabilities and measure of performances given for some values of arrival rate λ and the service parameters.

Approximate Transient Analysis of Queuing Networks by Decomposition based on Time-Inhomogeneous Markov Arrival Processes

We address the transient analysis of networks of queues with exponential service times. Such networks can easily have such a huge state space that their exact transient analysis is unfeasible. In this paper we propose an approximate transient analysis technique based on decomposing the queues of the network using a compact and approximate representation of the departure process of each queue. Namely, we apply time-inhomogeneous Markov arrival processes (IMAP) to describe the stream of clients leaving the queues. By doing so, the overall approximate model of the network is a time-inhomogeneous continuous time Markov chain (ICTMC) with significantly less number of states than there are in the original Markov chain. The proposed construction of the output IMAP of a queue is based on its transient state probabilities. We illustrate the approach first on a single M/M/1 queue and analyze the goodness of fitting of the departure process by numerical examples. Then we extend the approach to networks of queues and evaluate the precision of the resulting technique on several simple numerical examples by comparing the exact and the approximate transient probabilities of the queues.

Optimal Control of Service Parameter for a Perishable Inventory System with Service Facility, Postponed Demands and Finite Waiting Hall

This paper deals with the problem of optimally control service rates for a continuous review (s, S) inventory system with a service facility consisting of finite buffer(waiting hall) and a single server. The customers arrive according to a Poisson process. The customer’s demand is satisfied after an exponential service time. An arriving customer, who finds the buffer is full, enters into the pool of finite size or leaves the system according to a Bernoulli trial. The replenishment time of the order is distributed as exponential. The life time of each item in the inventory is assumed to be exponential distribution. Here we determine the service rates to be employed at each instant of time so that the long-run total expected cost rate is minimized. The problem is modelled as a semi-Markov decision problem. The stationary optimal policy is computed using linear programming algorithm and the results are illustrated numerically.

Analyzing rental vehicle threshold policies that consider expected waiting times for two customer classes

Vehicle rental providers, which operate in an uncertain environment, offer differentiated services to priority and non-priority customers. In this research, we study one such service differentiation strategy, a vehicle threshold policy, which is to hold vehicles for priority class customers in anticipation of their future arrivals. To consider the impact that vehicle threshold policies have on priority and non-priority customer waiting times, we model a rental depot as a multi-class non-work-conserving semi-open queue with stochastic inputs. To analyze the effect of vehicle rental period distributions, we identify and analyze the optimal threshold quantity for stationary customer arrivals with exponential and deterministic service time distributions. For non-stationary customer arrivals, we develop different threshold policies and analyze their performance using a detailed simulation model to conduct numerical experiments. We find that a maximum threshold policy is recommended when the average arrival rate of the priority customers is larger than the average arrival rate of the non-priority customers, and a stationary independent period by period policy is recommended when the average arrival rates of the customer classes are equal and the average best case utilization is less than 0.60. Finally, we analyze the impact that allowing priority customers to upgrade to a higher class of vehicles has on threshold policies.

An (s; Q) Inventory System with Positive Lead Time and Service Time Under N-Policy

Abtsrcat We consider an ( s, Q) inventory system of non-perishable items with exponential service time where N-policy is adopted during lead time. Ordering quantity is fixed, and is Q = S - s. We assume that arrival of demands is according to a Poisson process and lead time follows exponential distribution. N-policy is used in such a way that as and when the onhand inventory reaches the level s - N during a lead time, a local purchase of Q + N units is made by cancelling the replenishment order that is already placed. We obtain product-form solution for the state-probabilities. Several performance measures of the system are derived. Analysis of inventory cycle length is carried out. An explicit cost function is obtained and is analysed numerically. Numerical illustrations indicate that the cost function is strictly convex in N. Also it is numerically established that the total expected cost as a function of s, S and N is strictly convex.

Optimal control of noncollaborative servers in two‐stage tandem queueing systems

AbstractWe consider two‐stage tandem queueing systems with dedicated servers in each station and a flexible server that is trained to serve both stations. We assume no arrivals, exponential service times, and linear holding costs for jobs present in the system. We study the optimal dynamic assignment of servers to jobs assuming a noncollaborative work discipline with idling and preemptions allowed. For larger holding costs in the first station, we show that (i) nonidling policies are optimal and (ii) if the flexible server is not faster than the dedicated servers, the optimal server allocation strategy has a threshold‐type structure. For all other cases, we provide numerical results that support the optimality of threshold‐type policies. Our numerical experiments also indicate that when the flexible server is faster than the dedicated server of the second station, the optimal policy may have counterintuitive properties, which is not the case when a collaborative service discipline is assumed. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 435–446, 2014

An [formula omitted] queue where customers are served subject to a minimum violation of FCFS queue discipline

This article discusses the steady state analysis of the M/G/2 queuing system with two heterogeneous servers under new queue disciplines when the classical First Come First Served ‘(FCFS)’ queue discipline is to be violated. Customers are served either by server-I according to an exponential service time distribution with mean rate μ or by server-II with a general service time distribution B(t). Sequel to some objections raised in the literature on the use of the classical FCFS queue discipline in heterogeneous service systems, two alternative queue disciplines (Serial and Parallel) are considered in this work with the objective that if the FCFS is violated then the violation is a minimum in the long run. Using the embedded method under the serial queue discipline and the supplementary variable technique under the parallel queue discipline, we present an exact analysis of the steady state number of customers in the system and most importantly, the actual waiting time expectation of customers in the system. Our work shows that one can obtain all stationary probabilities and other vital measures for this queue under certain simple but realistic assumptions.

A resource-sharing game with relative priorities

Motivated by cloud-based computing resources operating with relative priorities, we investigate the strategic interaction between a fixed number of users sharing the capacity of a processor. Each user chooses a payment, which corresponds to his priority level, and submits jobs of variable sizes according to a stochastic process. These jobs have to be completed before some user-specific deadline. They are executed on the processor and receive a share of the capacity that is proportional to the priority level. The users’ goal is to choose priority levels so as to minimize their own payment, while guaranteeing that their jobs meet their deadlines. We fully characterize the solution of the game for two classes of users and exponential service times. For an arbitrary number of classes and general service times, we develop an approximation based on heavy-traffic and we characterize the solution of the game under the heavy-traffic assumption. Our experiments show that the approximate solution captures accurately the structure of the equilibrium in the original game.

A Skill Based Parallel Service System Under FCFS-ALIS — Steady State, Overloads, and Abandonments

We consider a queueing system with J parallel servers [Formula: see text], and with customer types 𝒞 = {a, b, …}. A bipartite graph G describes which pairs of server-customer types are compatible. We consider FCFS-ALIS policy: A server always picks the first, longest waiting compatible customer, and a customer is always assigned to the longest idle compatible server. We assume Poisson arrivals and server dependent exponential service times. We derive an explicit product-form expression for the stationary distribution of this system when service capacity is sufficient. We also calculate fluid limits of the system under overload, to show that local steady state exists. We distinguish the case of complete resource pooling when all the customers are served at the same rate by the pooled servers, and the case when the system has a unique decomposition into subsets of customer types, each of which is served at its own rate by a pooled subset of the servers. Finally, we discuss possible behavior of the system with generally distributed abandonments, under many server scaling. This paper complements and extends previous results of Kaplan, Caldentey and Weiss [18], and of Whitt and Talreja [34], as well as previous results of the authors [4, 35] on this topic.

The steady-state distribution of the [formula omitted] queue with a sinusoidal arrival rate function

The number of customers in a stable Mt/GI/n queue with a periodic arrival rate function and n servers has a proper steady-state limiting distribution if the initial place within the cycle is chosen uniformly at random. Insight is gained by examining the special case with infinitely many servers, exponential service times and a sinusoidal arrival rate function. Heavy-traffic limits help explain an unexpected bimodal form. The peakedness (ratio of the variance to the mean) can be used for approximations with finitely many servers.

Many server queuing problem with limited waiting space

In this paper the multi-channel queuing system with finite waiting space has been considered under poison arrival and exponential service time distributions. The Laplace transform of the time-dependent probability generating function has been obtained. In two and three channel cases probabilities have been derived explicate and known equilibrium results are shown to hold. At the end are given the tables for mean number of units in the system, with two and three channels, and the probability that an arriving unit on finding the system occupied is lost.

Optimal appointment scheduling with no-shows and exponential service time considering overtime work

Appointment systems are used by health clinics to manage access to service providers. In such systems, a specified number of patients are scheduled in advance, but certain patients may not arrive or ‘show up’ to their appointments. The existence of no-show behaviour influences both the operational cost of the clinics and the waiting time of the patients. In this paper, we determine an optimal schedule that takes no-show behaviour into account to determine the time intervals between patients under the framework of the individual-block/variable-interval rule for minimising the overall cost of the patient waiting time, the practitioner idle time and overtime. Under the condition that the service time of each patient is exponentially distributed, we compare the results with a schedule designed for the same expected number of patients in the absence of no-shows and analyse the effect on the system performance from the perspectives of day-length, expected workload, no-show probability, ratio of overtime costs and no-golf policy. We extend our results to an equally-spaced appointment system, which is commonly used in practice. Our results show that not only do no-shows greatly affect the system performance compared with an appointment system with the same expected workload without no-shows, but they also affect the optimal scheduling behaviours in the dome-shaped distribution. In addition, overtime cannot be eliminated completely even if the day length is adequate for all patients because of the stochastic characteristic of service time.

The Number Served in a Head Of The Line Priority Queue with General Service

In this paper, using the supplementary variable method, the head-of-the-line priority queueing system with general service time distributions have been studied to obtain an expression for the Laplace transform of the generating function of the joint probability distribution of the number of priority and non-priority units in the queue at time t, and the number of units served (including non-priority units) in time t. In particular cases, explicit solution for the model with exponential service time distributions have been obtained; expression for the joint distribution of the queue length and the number served in time t, for an M/G/1 queue has also been derived.

Many-server heavy-traffic limit for queues with time-varying parameters

A many-server heavy-traffic FCLT is proved for the $G_t/M/s_t+\mathit {GI}$ queueing model, having time-varying arrival rate and staffing, a general arrival process satisfying a FCLT, exponential service times and customer abandonment according to a general probability distribution. The FCLT provides theoretical support for the approximating deterministic fluid model the authors analyzed in a previous paper and a refined Gaussian process approximation, using variance formulas given here. The model is assumed to alternate between underloaded and overloaded intervals, with critical loading only at the isolated switching points. The proof is based on a recursive analysis of the system over these successive intervals, drawing heavily on previous results for infinite-server models. The FCLT requires careful treatment of the initial conditions for each interval.