Exponential Service Time Distribution Research Articles

An analysis of the equilibrium strategies for route-choosing customers in a two-station queueing system

We investigate a two-station queueing system where strategic customers must be sequentially serviced at both stations. We prove that an established property — that the distribution of the total time spent in a two-station system is independent of the chosen route when services times are exponentially distributed — is not a general one by providing a counterexample with deterministic service times. In doing so, we also prove a concomitant property — that any routing strategy is an equilibrium — is peculiar to a system with a particular assumption of an exponential service time distribution. Using simulations, we show that — depending on the distribution of service times — there can be (1) cases with three equilibria, (2) cases with one pure strategy equilibrium, and (3) cases with one mixed strategy equilibrium.

Transient Behavior of a Two-Phase Queuing System with a Limitation on the Total Queue Size

The transient mode of a two-phase queuing system with a Poisson input flow, exponential distribution of service time in each phase, and a limitation on the total buffer size of the two phases is considered. Nonstationary probabilities of system states are found using the Laplace transform. A numerical calculation and analysis of the system performance characteristics in transient mode with parameters corresponding to new-generation optical networks were carried out.

Determination of probabilistic-time characteristics when studying load properties in computer networks

The article is devoted to the determination of probabilistic-time characteristics when studying the properties of load in computer networks. The proposed approach to determining these characteristics is based on consideration of the probabilistic-time characteristics of a single-line queuing system with a geometric input flow and an exponential distribution of service time with a limited queue (g/M/1/N). The work takes into account that the flow of messages at the input of a subscriber station, hub or net-work controller serving a multipoint network with a “ring” or “tree” structure differs from Poisson. The proposed analysis method (G/M/1/N type system), called the parameter method, is illustrated with examples of various types of queuing systems.

Математическая модель расчета и анализа характеристик оптического коммутатора в переходном режиме

The article considers a transient behavior of an optical switch described using a single-channel finite buffer queuing system with Poisson input stream and exponential service time distribution. To solve the Kolmogorov equations system and find the probabilities of states in the transition mode, the apparatus of the Laplace transform was used. Analytical expressions are found for determining the probability of losses and the throughput of an optical switch at a given point in time. In a numerical example, the performance metrics of an optical switch with a buffer size of two packets are analyzed for different network loads in transient and stationary modes. The stationary performance metrics obtained using the proposed approach coincided with the known results, which confirms the correctness of the conclusions. The optical switch transient behavior analyses at different network load showed that with a decreasing service rate the transition mode time increases.

Pseudo Steady-State Period in Non-Stationary Infinite-Server Queue with State Dependent Arrival Intensity

An infinite-server queueing model with state-dependent arrival process and exponential distribution of service time is analyzed. It is assumed that the difference between the value of the arrival rate and total service rate becomes positive starting from a certain value of the number of customers in the system. In this paper, time until reaching this value by the number of customers in the system is called the pseudo steady-state period (PSSP). Distribution of duration of PSSP, its raw moments and its simple approximation under a certain scaling of the number of customers in the system are analyzed. Novelty of the considered problem consists of an arbitrary dependence of the rate of customer arrival on the current number of customers in the system and analysis of time until reaching from below a certain level by the number of customers in the system. The relevant existing papers focus on the analysis of time interval since exceeding a certain level until the number of customers goes down to this level (congestion period). Our main contribution consists of the derivation of a simple approximation of the considered time distribution by the exponential distribution. Numerical examples are presented, which confirm good quality of the proposed approximation.

Modified Erlang Loss System for Cognitive Wireless Networks

This paper considers a modified Erlang loss system for cognitive wireless networks and related applications. A primary user has pre-emptive priority over secondary users, and the primary customer is lost if upon arrival all the channels are used by other primary users. Secondary users cognitively use idle channels, and they can stay (either in an infinite buffer or in an orbit) in cases where idle channels are not available upon arrival or they are interrupted by primary users. While the infinite buffer model represents the case with zero sensing time, the infinite orbit model represents the case with positive sensing time. We obtain an explicit stability condition for the cases where arrival processes of primary users and secondary users follow Poisson processes, and their service times follow two distinct arbitrary distributions. The stability condition is insensitive to the service time distributions and implies the maximal throughout of secondary users. Moreover, we extend the stability analysis to the system with outgoing calls. For a special case of exponential service time distributions, we analyze the buffered system in depth to show the effect of parameters on the delay performance and the mean number of interruptions of secondary users. Our simulations for distributions rather than exponential reveal that the mean number of terminations for secondary users is less sensitive to the service time distribution of primary users.

An Analysis of a Stochastic ON-OFF Queueing Mobility Model for Software-Defined Vehicle Networks

We have recently witnessed a number of new software-defined paradigms of VANET in what is referred to as software-defined vehicle networks (SDVN). In order to evaluate the performance of these new proposals and architectures, analytical and simulation models are needed. In this paper, we propose an analytical model based on ON-OFF queueing networks under exponential and general service time distributions. The model can be used to evaluate the performance of SDVNs and takes into account the effect of mobility such as, hand overs, node turning ON/OFF, node going temporary out of coverage, and intermittent connections. This mobility effect was modelled as a queueing station with exponentially random ON-OFF service times, where traffic arrives according to a Poisson random process during the exponentially random ON period and the service time is exponentially distributed. However, during the OFF period the service time is exponentially distributed but with lower rates. We studied the ON-OFF queueing behaviour extensively for both finite-capacity and infinite-capacity queues. Three hypothetical SDVN scenarios were considered, taking into account the effect of mobility and the large number of connected nodes. Results were cross-validated with those obtained by a simulation model. These tools will be valuable for researchers interested in getting quantitative answers for their SDVN architectures.

Optimization of M/M/s/N Queueing Model with Reneging in a Fuzzy Environment

This paper deals with the study of multi-server queueing model in a fuzzy environment with imposition of reneging of customers. Entry of the customers in the system is assumed to be Poisson process and exponential service time distribution under first-come-first-served basis. Specific of this investigation is to derive the various fuzzy performance measures such as fuzzy queue length, fuzzy waiting time in queue, fuzzy response time and fuzzy optimal number of servers in explicit form for the finite capacity multi-server queueing system by using recursive method. For the validity of the model we have obtained the numerical illustrations in tabular form which shows that fuzzy-queue can be more realistic than crisp queue.

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.

Redundancy with Processor Sharing servers

The main motivation to investigate redundancy models comes from empirical evidence suggesting that redundancy can help improve the performance of real-world applications. Under redundancy, a job that arrives to this system is dispatched to d servers uniformly chosen at random in order to benefit from the variability of the length of these queues. As soon as one of the copies finishes service, the job (and its copies) is removed from the system, and as a consequence, a job's delay is given by the minimum delay among the servers its copies are sent to. Most of the literature on performance evaluation of redundancy systems has been carried out when First Come First Served (FCFS) is implemented in the servers. In particular, for exponential service time distributions, Gardner et al. [4, 5] and Bonald and Comte [2] show that the stability region is not reduced due to adding redundant copies. In this extended abstract, we focus instead on Processor Sharing (PS) service policy and study how redundancy impacts the stability condition. In particular, we aim to study the impact that the correlation structure of the copies has on the performance of the redundancy-d model. In a recent paper, Gardner et al. [3] showed that the assumption of independent and identically distributed (i.i.d.) copies, can be unrealistic, and that it might lead to theoretical results that do not reflect the results of replication schemes in real-life computer systems. We consider the two extreme cases of correlation; (i) the copies are i.i.d. (ii) the copies of a job are exact replicas (identical copies). We observe that the stability condition strongly depends on the correlation structure, as well as on the number of redundant copies.

Queueing analysis for operations modeling in port logistics

Purpose The use of queueing network models was stimulated by the appearance (1975) of the exact product form solution of a class of open, closed and mixed queueing networks obeying the local balance principle and solved, a few years later, by the popular mean value analysis algorithm (1980). Since then, research efforts have been produced to approximate solutions for non-exponential services and non-pure random mechanisms in customer processing and routing. The purpose of this paper is to examine the suitability of modeling choices and solution approaches consolidated in other domains with respect to two key logistic processes in container terminals. Design/methodology/approach In particular, the analytical solution of queueing networks is assessed for the vessel arrival-departure process and the container internal transfer process with respect to a real terminal of pure transshipment. Findings Numerical experiments show the extent to which a decomposition-based approximation, under fixed or state-dependent arrival rates, may be suitable for the approximate analysis of the queueing network models. Research limitations/implications The limitation of adopting exponential service time distributions and Poisson flows is highlighted. Practical implications Comparisons with a simulation-based solution deliver numerical evidence on the companion use of simulation in the daily practice of managing operations in a finite-time horizon under complex policies. Originality/value Discussion of some open modeling issues and encouraging results provide some guidelines on future research efforts and/or suitable adaption to container terminal logistics of the large body of techniques and algorithms available nowadays for supporting long-run decisions.

Unified killing mechanism in a single server queue with renewal input

Queueing systems experienced in real-life situations are very often influenced by negative arrivals which are independent of service process and cause the elimination of jobs from the system. Such a scenario occurs in computer network and telecommunication systems where an attack by a malicious virus results in the removal of some or all data files from the system. Along this direction many authors have proposed various killing processes in the past. This paper unifies different killing mechanisms into the classical single server queue having infinite capacity, where arrival occurs as renewal process with exponential service time distribution. The system is assumed to be affected by negative customers as well as disasters. The model is investigated in steady-state in a very simple and elegant way by means of supplementary variable and difference equation technique. The distribution of system-content for the positive customers is derived in an explicit form at pre-arrival and random epochs. The influence of different parameters on the system performance are also examined.

재시도 준개방형 대기행렬네트워크 분석

A semi-open queueing network is considered. The network of an arbitrary topology consists of a finite number of nodes operation of which is modeled by single-server queueing systems with a buffer and an exponential service time distribution. Customers arrive at the network according to a Markovian arrival process. The total number of customers, which can be processed simultaneously, is restricted by a fixed in advance threshold. If the number of customers in the network at a new customer arrival moment is less than the threshold, this customer is admitted for service. Otherwise, the customer temporarily leaves the system (it is said that the customer joins the orbit) and retries to get access to the network after the random time intervals duration of which has an exponential distribution. The capacity of the orbit is infinite. After the admission to the network, the customer starts sequential service in a random number of nodes. The choice of the first and the subsequent nodes for service is made stochastically according to a fixed stochastic vector and a transition probability matrix. The behavior of the network is described by the continuous-time Markov chain that belongs to the class of asymptotically quasi-Toeplitz Markov chains. This allows to derive a transparent condition for the ergodicity of the chain and compute its steady-state distribution. Expressions for computation of various performance measures of the network are derived. Numerical results are presented. The model can be applied, e.g., for analysis and optimization of the operation of logistics and manufacturing systems, and the wireless telecommunication networks.

Performance analysis and stability of multiclass orbit queue with constant retrial rates and balking

In this paper, we consider a single-server retrial model with multiple classes of customers. Arrival of customers follow independent Poisson rule. A new customer, facing a busy server upon his arrival, may join the corresponding (class-dependent) orbit queue with a class-dependent probability, or leaves the system forever (balks). The orbit queues follow constant retrial rate discipline, that is, only one (oldest) orbital customer of each orbit queue makes attempts to occupy the server, in a gap of class-dependent exponential times. Within each class, service times are assumed to be independent and identically distributed (iid). We show that this setting generalizes the so-called two-way communication systems.This multiclass system with general service time distributions is analysed using regenerative approach. Necessary and sufficient stability conditions, as well as some explicit expressions for the basic steady-state probabilities, are obtained. A restricted, two-way communication model with exponential service time distributions, is analysed by matrix-analytic method. Moreover, we combine both methods to efficiently derive explicit solutions for the restricted model.An extensive simulation analysis is performed to gain deep insight into the model stability and performance. It is shown that both the simulated and exact results agree on some important measures for which analytical expressions are available, and hence establish the validity of our theoretical treatment. We numerically study the sophisticated dependence structure of the model to uncover the orbits interaction. We give further details and intuitive explanation for the system performance which complements the derived explicit expressions.

Modeling Curbside Parking as a Network of Finite Capacity Queues

Paid curbside parking can be advantageously modeled as a network of interdependent queues. To this end, we introduce methods for analyzing a special class of networks of finite capacity queues where drivers arrive from an exogenous source, join the queue if there is an available parking space, or continue to search at an adjacent queue for an available space. Furthermore, we apply this model to estimate the proportion of drivers cruising in the neighborhood of Belltown, Seattle, WA, USA. Using occupancy approximated by parking transaction data, we estimate the percentage of drivers cruising for curbside parking by comparing the rate of drivers unable to find parking to bulk through-traffic measurement data. We find percentages of up to 50% for a Belltown's 1st Ave. depending on the time, day, and direction of travel. We then calculate a per vehicle travel-time cost to social welfare incurred by this proportion: upward of a 10% increase in travel time to all drivers along 1st Ave. Last, we introduce a simulation tool and test assumptions made when estimating interesting performance metrics like the probability of a block-face being full. Our results suggest that while assuming exponential service time distributions is not justified, mean rate solutions under a Markovian relaxation of the problem is comparable to service times representative of parking transaction data in simulation.

Analysis of a semi-open queueing network with Markovian arrival process

A semi-open queueing network having a finite number of nodes is considered. The nodes are modeled by single-server queueing systems with a finite buffer and an exponential service time distribution. Customers arrive to the network according to a Markovian arrival process. The number of customers, which can be processed in the network simultaneously, is restricted by a threshold. If the number of customers in the network is less than this threshold, when a new customer arrives, the customer is processed in the network. Choice of the first and the subsequent nodes for service is performed randomly according to a fixed stochastic vector and a transition probability matrix. If the number of customers in the network at the customer arrival epoch is equal to the threshold, the customer is queued into an input buffer with an infinite capacity. Customers in the input buffer are impatient. The stationary behavior of network states is analyzed. The Laplace–Stieltjes transform of the distribution of the customer’s waiting time in the input buffer is obtained. Expressions for computing performance measures of the network are derived. Numerical results are presented. The model is suitable, e.g., for analysis and optimization of wireless telecommunication networks and manufacturing systems with a finite number of machines and workers.

Multi server machine repair problem with unreliable server and two types of spares under asynchronous vacation policy

In many industrial situations, machines are subjected to random breakdowns. In this paper, we consider the asynchronous vacation policy for the multi-server machine repair problem with server breakdown and two types of spares. According to this policy, when a certain number of servers become idle at a service completion instant, the servers will take an asynchronous vacation of random length together. If the servers find no waiting failed unit in the system and the number of servers on vacation is less than the total number of servers, then these servers can take vacation individually and continue to take vacation if the vacation condition is satisfied. Each server is subject to breakdowns in working state only and can be repaired according to exponential service time distribution. Matrix geometric method is used to obtain the queue size distribution. Various performance measures viz. the expected number of the idle servers, busy servers and vacationing servers, etc. are determined. The sensitivity analysis is made to examine the effect of parameters on various performance measures.

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 [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.

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.