Single Server Queueing Model Research Articles

Analysis of a Queueing Model with Batch Markovian Arrival Process and General Distribution for Group Clearance.

In this paper we consider a single server queueing model with under general bulk service rule with infinite upper bound on the batch size which we call group clearance. The arrivals occur according to a batch Markovian point process and the services are generally distributed. The customers arriving after the service initiation cannot enter the ongoing service. The service time is independent on the batch size. First, we employ the classical embedded Markov renewal process approach to study the model. Secondly, under the assumption that the services are of phase type, we study the model as a continuous-time Markov chain whose generator has a very special structure. Using matrix-analytic methods we study the model in steady-state and discuss some special cases of the model as well as representative numerical examples covering a wide range of service time distributions such as constant, uniform, Weibull, and phase type.

(M, MAP)/(PH, PH)/1 Queue with Non-preemptive Priority and Working Vacation Under N-Policy

In this paper we consider two single server queueing models with non-preemptive priority and working vacation under two distinct N-policies. High priority (type I) customers are served even in vacation mode whereas low priority (type II) customers are served only when the server comes to normal mode of service. Type I customers have only a limited waiting space L whereas type II customers have unlimited capacity. The two distinct N-policies are as described below: In model I, while service of type I customers are in progress in vacation mode (working vacation), if the number of such customers present in the system hits N ($$\le L$$) or the vacation timer (clock) expires, whichever occurs first, the server is switched on to normal mode. In model II, switching the server to normal mode from vacation mode occurs as soon as the accumulated number (those served out plus those present in the system) of type I customers during that working vacation hits N or the vacation timer expires, whichever occurs first. Type I customers arrive according to a Poisson process whereas type II customer’s arrival is governed by Markovian Arrival Process. Service time of type I and type II customers follow distinct phase type distributions. At a service completion epoch, finding the system empty, server takes an exponentially distributed working vacation. During working vacation, type I customers are served at a reduced rate. On vacation expiration, the service of the type I customer already in service, will start from the beginning in the normal mode of service. We analyze these models in steady state to compute the distribution of duration of service time continuously in slow mode, expected number of returns to 0 type I customer state, starting from 0 type I customer state during vacation mode of service before the arrival of a type II customer, the distribution of a p-cycle in normal mode, LSTs of busy cycle, busy period of type I customers generated during the service time of a type II customer and LSTs of waiting time distributions of type I and type II customers. We compare these models in steady state by numerical experiments to identify the superior model.

Moderate deviation for maximum likelihood estimators from single server queues

Consider a single server queueing model which is observed over a continuous time interval (0,T], where T is determined by a suitable stopping rule. Let θ be the unknown parameter for the arrival process and hat {theta }_{T} be the maximum likelihood estimator of θ. The main goal of this paper is to obtain a moderate deviation result of the maximum likelihood estimator for the single server queueing model under certain regular conditions.

Heavy outgoing call asymptotics for retrial queue with two way communication and multiple types of outgoing calls

In this paper, we consider a single server queueing model M |M |1|N with two types of calls: incoming calls and outgoing calls, where incoming calls arrive at the server according to a Poisson process. Upon arrival, an incoming call immediately occupies the server if it is idle or joins an orbit if the server is busy. From the orbit, an incoming call retries to occupy the server and behaves the same as a fresh incoming call. The server makes an outgoing calls after an exponentially distributed idle time. It can be interpreted as that outgoing calls arrive at the server according to a Poisson process. There are N types of outgoing calls whose durations follow N distinct exponential distributions. Our contribution is to derive the asymptotics of the number of incoming calls in retrial queue under the conditions of high rates of making outgoing calls and low rates of service time of each type of outgoing calls. Based on the obtained asymptotics, we have built the approximations of the probability distribution of the number of incoming calls in the system.

Multimodularity in the Stochastic Appointment Scheduling Problem with Discrete Arrival Epochs

We address the problem of designing appointment scheduling strategies in a stochastic environment accounting for patient no-shows, nonpunctuality, general stochastic service times, and unscheduled emergency walk-ins. A good appointment schedule seeks to help outpatient clinics to utilize their resources efficiently while containing patients’ waiting times. The task of identifying an optimal schedule is modeled as a nonlinear integer program, where the objective function is the outcome of stochastic analysis in transient state. We maintain the discrete nature of the appointment scheduling problem by considering arrival epochs with discrete supports. By looking at discrete-time snapshots of the random evolution of a single-server queueing model, we characterize probabilistically the system’s workload over time as a function of an appointment schedule, and we derive recursive expressions for the performance measures of interest. Subsequently, we unfold discrete convexity properties of the optimization problem. We prove that under general conditions the objective function is supermodular and componentwise convex. Under assumptions on patient punctuality, we prove that the optimal scheduling strategy minimizes a multimodular function, a property which guarantees that a locally optimal schedule is also globally optimal. The size of the local neighborhood, however, grows exponentially with the dimension of the problem. To the best of our knowledge, this study is the first to develop and implement an algorithm for minimizing locally a multimodular function over nonnegative integer vectors via submodular set-function minimization over ring families, a task that can be performed in polynomial time. This paper was accepted by David Simchi-Levi, operations management.

Bayesian inference and prediction in single server M/M/1 queuing model based on queue length

This paper is concerned with the problem of estimating traffic intensity, ρ for single server queuing model in which inter-arrival and service times are exponentially distributed (Markovian) using data on queue size (number of customers present in the queue) observed at any random point of time. Here, it is assumed that q is unknown but random quantity. Bayes estimator of ρ are derived under squared error loss function assuming two forms of prior information on ρ. The performance of the proposed Bayes estimators is compared with that of the corresponding classical version estimator based on maximum likelihood principle. The model comparison criterion based on Bayes factor is used to select a suitable prior for Bayesian analysis.

A single server queueing model with self-generation of priorities, customer induced interruption and retrial of customers

A single server queueing model with self-generation of priorities, customer induced interruption and retrial of customers

Modified Bessel series solution of the single server queueing model with feedback

In multi-access systems, scheduling mechanism often requires a proper feedback policy. In this paper, the direct and simple transient solution technique for the state of the system in a single server/processor Markovian queueing model with feedback is presented using the modified Bessel function of the second kind. This technique appears to economise in algebra. The expression for a time-dependent measure of effectiveness such as an expected number of the customers in the system is also derived. We demonstrate how fast the state probabilities tend to their equilibrium limits when it exists. The sensitivity of the state of the system and expected number of the customers in the system has been also analysed and the results are depicted in the tables and graphs.

Routing Algorithm for Fastest Path – a Queuing Approach

Finding a fastest route for data packets is a complex process as far as large scale network is considered. Fastest path helps to minimize the delay, data loss and in turn minimizes the total cost of the network. Dijkstra’s algorithm is one of the most commonly used algorithms for finding the shortest path from one source node to another destination node, which pertains to positive weights only. In this paper, Dijkstra’s Algorithm combined with the features of single server queuing model is presented to find the fastest route between any two given nodes. In this improvised algorithm, every edge along with its end vertices is considered as M/M/1 queuing model. The Waiting time of a packet, number of packets in the system and the probability that the queue length exceeds packets are considered along with the weight of the particular link, for the calculation of the fastest route.

Multi-threshold control by a single-server queuing model with a service rate depending on the amount of harvested energy

We consider a single-server queuing model with energy harvesting. Customers and energy units arrive in the marked Markovian arrival flow. Buffers for customers and energy are finite. There are several service modes distinguished by the number of required units of energy, the service rate and the quality of service. The choice of the service mode for the next customer depends on the current number of energy units in the system. The quality of system operation is assumed to be defined by a customer’s loss probability. Loss can occur due to buffer overflow, low-quality service or impatience during the waiting time. The control strategy is of a multi-threshold type with thresholds defined by the amount of available energy. Under the fixed values of the thresholds, the behaviour of the system is described by a multi-dimensional Markov chain. The generator of this chain is derived. The main performance measures are calculated. The distribution of the waiting time is derived in explicit form. A numerical study of the model via extensive experiments that provide insight into the quantitative behaviour of the system is presented.

Optimizing buffer size for the retrial queue: two state space collapse results in heavy traffic

We study a single server queueing model with admission control and retrials. In the heavy traffic limit, the main queue and retrial queue lengths jointly converge to a degenerate two-dimensional diffusion process. When this model is considered with holding and rejection costs, formal limits lead to a free boundary curve that determines a threshold on the main queue length as a function of the retrial queue length, above which arrivals must be rejected. However, it is known to be a notoriously difficult problem to characterize this curve. We aim instead at optimizing the threshold on the main queue length independently of the retrial queue length. Our main result shows that in the small and large retrial rate limits, this problem is governed by the Harrison–Taksar free boundary problem, which is a Bellman equation in which the free boundary consists of a single point. We derive the asymptotically optimal buffer size in these two extreme cases, as the scaling parameter and the retrial rate approach their limits.

Analysis of a queueing system in random environment with an unreliable server and geometric abandonments

This paper studies a single server queueing model in a multi-phase random environment with server breakdowns and geometric abandonments, where server breakdowns only occur while the server is in operation. At a server breakdown instant (i.e., an abandonment opportunity epoch), all present customers adopt the so-called geometric abandonments, that is, the customers decide sequentially whether they will leave the system or not. In the meantime, the server abandons the service and a repair process starts immediately. After the server is repaired, the server resumes its service, and the system enters into the operative phaseiwith probabilityqi,i= 1, 2, …,d. Using probability generating functions and matrix geometric approach, we obtain the steady state distribution and various performance measures. In addition, some numerical examples are presented to show the impact of parameters on the performance measures.

ANALISIS ANTRIAN SINGLE CHANNEL SINGLE SERVER DENGAN LAYANAN BERKELOMPOK PADA KONEKSI INTERNET DI UNIVERSITAS ISLAM INDONESIA

Internet is very important to support students in getting knowledge. However, internet access in college is sometimes not proportional to the number of users. Current speed of wireless network connection in Universitas Islam Indonesia (UII), named UIIAccess, is also not proportional to the number of users. Based on these problems, this research was conducted to study UIIAccess performance in providing internet service. The analysis used is single channel single server queuing model with bulk service. The result shows that traffic intensity of the system is 0.9694 indicating that the intensity of UIIAccess service is very solid. The queue performance shows that the average number of users in the system and waiting in the queue are respectively 32 and 11 users per minute, the average time spent by a user in the system and in the queue are consecutively 10.45574 and 3.63663 minutes. It indicates that the queue on UIIAccess is very crowded because of the imbalance between access speed and the number of users

Transient performance analysis of a single server queuing model with retention of reneging customers

In this paper, we study a single server queuing model with retention of reneging customers. The transient solution of the model is derived using probability generating function technique. The time-dependent mean and variance of the model are also obtained. Some important special cases of the model are derived and discussed. Finally, based on the numerical example, the transient performance analysis of the model is performed.

An Unreliable Quorum Queueing System with Single and Multiple Vacations

This paper contributes to the literature of single server queueing models with a vacationing server. We have incorporated many features for a better control over the system. The server implements the N-policy, takes both single and multiple vacations, and is subject to breakdowns. The embedded Markov chain technique is used to obtain the pgf of the system size at a service completion epoch in the steady-state. The semi-regenerative technique is used to obtain the pgf of the system size at an arbitrary instant of time in the steady-state.

An optimization model to determine appointment scheduling window for an outpatient clinic with patient no-shows.

This paper investigates appointment scheduling for an outpatient department in West China Hospital (WCH), one of the largest single point of access hospitals in the world. Our pilot data analysis shows that the appointment system at WCH can be improved through leveraging the scheduling window (i.e., the number of days in advance a patient makes an appointment for future services). To gain full insight into this strategy, our study considers two cases, based on if patients are willing to wait for scheduled appointments or not. We developed a stylized single server queueing model to find optimal scheduling windows. Results show that, when patients are less sensitive to time delay (i.e., patients will wait for scheduled services), levering scheduling windows is not effective to minimize the total cost per day of the appointment system. In contrast, when patients are sensitive to time delay (i.e., patients may find services elsewhere), then our model considers the potential cost of physician idle time. The modeling results indicate that the total cost per day is relatively sensitive to the magnitude of scheduling window. Thus, adopting a proper scheduling window is very important. In addition, our study proves that the cost functions of both cases are quasi-concave, which are also validated by actual data drawn from the Healthcare Information System at WCH. A comparison of numerical results between two cases is made to draw further managerial insights into scheduling policies for WCH. Discussion of our findings and research limitations are also provided.

Transient analysis of M/M/1 queue with working vacation, heterogeneous service and customers’ impatience

This paper studies the time-dependent behavior of a single server queueing model with slow service during single and multiple working vacations, and customers’ impatience due to slow service. The time-dependent system size probabilities for the underlying model are obtained in closed form. Further, time-dependent mean and variance are deduced. Finally, numerical illustrations are shown to visualize the system effect for various values of parameters.

English

One of the greatest challenges facing every society today is crime control and management. It could be as little as pick–pocketing, or human trafficking, or even as deadly as terrorism. As criminal’s perfect ways of avoiding being detected, Law Enforcement Agencies (LEAs) must adopt innovative ways on crime prevention and control. This research applies Queuing Network (QN) model to evaluation the performance of LEAs on crime information management. The QN comprise two queuing theory models; single server (M/M/1) and multiple server (M/M/m) queuing models. To implement this model, crime data was collected from Eleme (2012) and Nsukka (2013) police stations with a table format to capture the timing such as case arrival time, service time (investigation and handling time) and termination time. The model was implemented using PHP programming language Excel application was used to plot some graphs to observe the system’s behaviour. The case timing captured (input data) were used to calculate the queuing theory performance measures (model parameters). Results from the analysis shows how many cases were handled by how many staff members in a specified period of time. This model will make LEA’s crime management system visible at different levels (local, state and federal) to both government, LEA’s admins and general public. Government can assess the LEA’s performance at any time. Use of this model will improve LEA’s productivity and public security as well. Key words: Crime management, information system, law enforcement agency, performance evaluation, queueing network, single server queuing model, multiple server queueing model.

Performance analysis of an improved dynamic power management model in wireless sensor node

The energy consumption in the wireless sensor networks is a very critical issue which attracts immediate attention for the sake of the growing demand of the billion dollar market in future. The Dynamic Power Management (DPM) technique is a way of controlling and saving the energy usage in a sensor node. Previously, researchers have proposed lifetime improving stochastic models for wireless sensor networks and limited work has been done focusing on the wireless sensor node. This paper proposes an analyser based Semi-Markov model for DPM in the event-driven sensor node. The power consumption comparison with previously proposed models without this analyser shows the analyser significant contributes to lifetime improvement. The improved model is more power efficient, presents how the DPM model observes the input event arrival and power states of the sensor node components, and then dynamically manages the power consumption of the overall system. Further, to observe the effect of event arrival, missed events, waiting time, processor utilization on the power consumption and lifetime, the proposed DPM system with the single server queuing model is developed.

A Queuing model for Dealing with Patients with Severe Disease

This paper suggests a proposed single server queueing model for severe diseases especially in Out-patient Department. The Outpatient Department of a hospital is visited by patients of all types ofdisease. Some of these diseases require immediate medical attention as severe complications may ariseif treatment is delayed. The goal of the study was to develop a queueing model considering patientswith severe disease and to study the improvement in the service time using the model. The singleserver queueing model was modied and analyzed. The eciency of the model was tested by usingoutpatient medical service, arrivals and departure of patients over a period of one year of a localhospital in Guwahati. The result indicated the average outpatient medical service response times forservice improve over the general model.