Single Queue Research Articles

MRSPN analysis of Semi-Markovian finite source retrial queues

In this paper, the analysis of Semi-Markovian single server retrial queues by means of Markov Regenerative Stochastic Petri Nets (MRSPN) is considered. We propose MRSPN models for the two retrial queues M/G/1/N/N and M/G/1/N/N with orbital search. By inspecting the reduced reachability graph of both MRSPN models, the qualitative analysis is obtained. The quantitative analysis is carried out after constructing their one step transition probability matrix and computing the steady state probability distribution of each tangible marking. As an example, the queue $$M/Hypo_{2}/1/2/2$$ is treated in order to illustrate the functionality of the MRSPN approach. The exact performance measures (mean number of customers in the system, mean response time, mean waiting time,...) are computed for different parameters of the two systems by an algorithm elaborated in Matlab environment.

Efficient fairness using Early Discriminative Dropping

The queue management scheme in a router is critical for computer networks because it not only controls the fairness of bandwidth sharing among competing flows, but also contributes to enhancing the efficiency of congestion control algorithms. In this paper, a new queue management scheme, Early Discriminative Dropping (EDD), is proposed to make routers fairer. EDD employs a single queue. In addition, this scheme consists of three main components. Firstly, a threshold estimator evaluates control thresholds based on queue status and the number of accepted packets. Secondly, a quota estimator allocates quotas, which prevent arriving packets of non-aggressive flows from being discarded during congested traffic. Finally, a simple packet-dropping algorithm determines acceptance or discarding of the arriving packet according to current control threshold and available quotas. Using computer simulations, we show that the EDD scheme demonstrates robust and excellent fairness compared with Rotating Preference Queues, Core-stateless Fair Queueing, Dynamic Detection and Expulsion, CHOose and Keep for responsive flows, CHOose and Kill for unresponsive flows and First In First Out under various network topologies and traffic conditions. EDD has a complexity of O(1). Hence, it is feasible for deployment in computer networks where high-performance and high-speed routers exist.

Scaling limits for single server retrial queues with two-way communication

This paper studies M/G/1 retrial queues in which there are two arrival flows, i.e., incoming calls made by regular customers and outgoing calls made by the server in idle time. The stationary analysis of this system has been carried out in a recent paper by Artalejo and Phung-Duc (Appl Math Model 37(4):1811–1822, 2013). In this paper, we obtain a decomposition property where we prove that the queue length is decomposed into the sum of three independent random variables with clear physical meaning. We then derive scaling limits for the queue length distribution under some extreme conditions (i) heavy traffic, (ii) slow retrials and (iii) instantaneous connection to outgoing calls. Furthermore, we also investigate the convergence of our model to that without outgoing calls.

Single server retrial queue with group admission of customers

We consider a retrial queueing system with a single server and novel customer׳s admission discipline. The input flow is described by a Markov Arrival Process. If an arriving customer meets the server providing the service, it goes to the orbit and repeats attempts to get service in random time intervals whose duration has exponential distribution with parameter dependent on the customers number in orbit. Server operates as follows. After a service completion epoch, customers admission interval starts. Duration of this interval has phase type distribution. During this interval, primary customers and customers from the orbit are accepted to the pool of customers which will get service after the admission interval. Capacity of this pool is limited and after the moment when the pool becomes full before completion of admission interval all arriving customers move to the orbit. After completion of an admission interval, all customers in the pool are served simultaneously by the server during the time having phase type distribution depending on the customers number in the pool. Using results known for Asymptotically Quasi-Toeplitz Markov Chains, we derive stability condition of the system, compute the stationary distribution of the system states, derive formulas for the main performance measures and numerically show advantages of the considered customer׳s admission discipline (higher throughput, smaller average number of customers in the system, higher probability to get a service without visiting the orbit) in case of proper choice of the capacity of the pool and the admission period duration.

Real-time queue estimation model development for uninterrupted freeway flow based on shockwave analysis

In this study, the authors developed a time-space discrete macroscopic model based on the shockwave theory for real-time queue estimation in uninterrupted freeway flow, using fixed-location loop detector data. After investigating the queue characteristics both at an active bottleneck and within a variable speed limit control case, the proposed model was applied to these two cases on Whitemud Drive, a major freeway corridor in Edmonton, Alberta, Canada. Modified Highway Capacity Manual–based methods were used to determine queue density in uninterrupted freeway flow. The effect of time interval size on queue estimation was studied, as loop detector data acquisition frequencies may differ. It was found that the proposed model accurately estimates real-time queue length independent of the time interval. Multiple single queues were implemented in a calibrated VISSIM 5.3 micro-simulation model to perform the validation task. The study is a helpful foundation for future active traffic management strategy development and improvement.

Analysis of Hybrid Buffering and Retransmission in OBS Networks.

Burst contention is a major problem in the Optical Burst Switching (OBS) networks. Due to inadequate contention resolution techniques, the burst loss is prominent in OBS. In order to resolve contention fiber delay lines, wavelength converters, deflection routing, burst segmentation, and retransmission are used. Each one has its own limitations. In this paper, a new hybrid scheme is proposed which combines buffering and retransmission, which increases the mean number of bursts processed in the system. In this hybrid method, retransmission with controllable arrival and uncontrollable arrival is analyzed. Normally all the bursts reach the first hop and few of them go for second hop to reach destination. After all the bursts reach the destination the server may go for maintenance activity or wait for the arrival of next burst. We model it as a batch arrival single server retrial queue with buffer. Numerical results are analyzed to show the mean number of bursts processed in the system with uncontrollable arrival and controllable arrivals.

Analysis of a single server batch arrival retrial queueing system with modified vacations and N-policy

In this paper, a batch arrival single server retrial queue with modified vacations under N -policy is considered. If an arriving batch of customers finds the server busy or on vacation, then the entire batch joins the orbit in order to seek the service again. Otherwise, one customer from the arriving batch receives the service, while the rest joins the orbit. The customers in the orbit will try for service one by one when the server is idle with a classical retrial policy with the retrial rate ‘jv ’, where ‘j ’ is the size of the orbit. At a service completion epoch, if the number of customers in the orbit is zero, then the server leaves for a secondary job (vacation) of random length. At a vacation completion epoch, if the orbit size is at least N , then the server remains in the system to render service for the primary customers or orbital customers. On the other hand, if the number of customers in the orbit is less than ‘N ’ at a vacation completion epoch, the server avails multiple vacations subject to maximum ‘M ’ repeated vacations. After availing ‘M ’ consecutive vacations, the server returns to the system to render service irrespective of the orbit size. The model is studied using supplementary variable technique. For the proposed queueing system, the probability generating function of the steady state queue size distribution at an arbitrary time is obtained. Various performance measures are derived. A cost model for the queueing system is developed. Numerical illustration is provided.

Retrial queueing system with balking, optional service and vacation

In this study, we propose a single server retrial queueing system with balking, second optional service and single vacation. At the arrival epoch, if the server is busy, the arriving job join the orbit or balks the system whereas if the server is free, then the arriving job starts its service immediately. For each job, the server provides two phases of service. All the jobs demand the first essential service, whereas only some of the jobs demand for the second optional service. If the system is empty, then the server becomes inactive and begins a single vacation. If server comes back from the vacation, it does not go for another vacation even if the system is still empty at that time. The steady state distributions of the server state and the number of jobs in the orbit are obtained along with other performance measures. The effects of various parameters on the system performance are analyzed numerically. A general decomposition law for this retrial queueing system is established.

On a retrial queueing model with single/batch service and search of customers from the orbit

A single server retrial queueing system, where customers arrive according to a batch Poisson process and are served either in single or as a batch, is considered here. An arriving batch, finding the server busy, enters an orbit. Otherwise, one customer, a few customers, or all customers from the arriving batch, depending on if the batch size exceeds a threshold value or not, enter service immediately, while the rest join the orbit. Customers from the orbit try to reach the server subsequently with the inter-retrial times, exponentially distributed. Additionally, at each service completion epoch, one of the two types of search mechanisms say, type I and type II search, to bring the orbital customers to service, is switched on—type I search when the orbit size is less than the threshold value and type II search otherwise. This means that, while the server is idle, a competition takes place among primary customers, customers who come by retrial and by one of the two types of search as the case may be. A type I search selects a single customer whereas a type II search takes a batch of customers from the orbit. In the case of primary customers and those who come by type II search, maximum size of the batch taken into service is restricted to a pre-assigned value. Both single and batch service are assumed to be arbitrarily distributed with different distributions, which are independent of each other. Steady state analysis is performed. Some important system descriptors are computed algorithmically and numerical illustrations are provided.

Two-way communication retrial queues with multiple types of outgoing calls

In this paper, we start with single server Markovian retrial queues with multiple types of outgoing calls. Incoming calls arrive at system according to a Poisson process. Service times of incoming calls follow the exponential distribution. Incoming calls that find the server busy upon arrival join an orbit and retry after some exponentially distributed time. On the other hand, the server makes an outgoing call after some exponentially distributed idle time. We assume that there are multiple types of outgoing calls whose durations follow distinct exponential distributions. For this model, we obtain explicit expressions for the joint stationary distribution of the number of calls in the orbit and the state of the server via the generating function approach. We also obtain simple asymptotic and recursive formulae for the joint stationary distribution. We show a stochastic decomposition property where we prove that the number of incoming calls in the system (server and orbit) can be decomposed into the sum of three independent random variables which have a clear physical meaning. We then consider the multiserver model for which we obtain the stability condition and derive some exact formulae by mean value analysis. Finally, we extend the single server model to the case where service time distribution of incoming calls and that of each type of outgoing calls are arbitrary. For this case, we obtain explicit expressions for the partial generating functions and recursive formulae for the joint stationary distribution of the server’s state and the number of calls in the orbit.

Single Queue Attended by Alternative Servers with Changeover Times

A single queue is serviced by two servers alternately, in alternate busy periods. A changeover time is required whenever the servers are replaced. In this model, the changeover time of a server is initiated as soon as one server completes the services of the units waiting before him namely at the end of his busy period. It may be noted that the empty state of the system is never reached. A comparison of this model with the model with empty state has been done.

Analysis of a discrete-time working vacation queue with balking

This paper investigates a discrete-time single server finite buffer queue with single working vacation wherein the customers decide either to join the queue or balk. The inter-arrival times, service times and vacation times are assumed to be independent and geometrically distributed. Using matrix method, the steady-state probabilities are obtained. Various performance measures of the model and average sojourn time are presented. An optimization problem is considered using particle swarm optimization (PSO) through a cost function. Further, maximum likelihood estimate (MLE) for the balking parameter is obtained. Numerical results are presented in the form of tables and graphs to show the effect of parameters on the system performance.

M/M/1 Retrial queueing system with negative arrival under non-preemptive priority service

Consider a single server retrial queueing system with negative arrival under non-pre-emptive priority service in which three types of customers arrive in a poisson process with arrival rate λ1 for low priority customers and λ2 for high priority customers and λ3 for negative arrival. Low and high priority customers are identified as primary calls. The service times follow an exponential distribution with parameters μ1 and μ2 for low and high priority customers. The retrial and negative arrivals are introduced for low priority customers only. Gelenbe (1991) has introduced a new class of queueing processes in which customers are either positive or negative. Positive means a regular customer who is treated in the usual way by a server. Negative customers have the effect of deleting some customer in the queue. In the simplest version, a negative arrival removes an ordinary positive customer or a random batch of positive customers according to some strategy. It is noted that the existence of a flow of negative arrivals provides a control mechanismto control excessive congestion at the retrial group and also assume that the negative customers only act when the server is busy. Let K be the maximumnumber of waiting spaces for high priority customers in front of the service station. The high priorities customers will be governed by the Non-preemptive priority service. The access from the orbit to the service facility is governed by the classical retrial policy. This model is solved by using Matrix geometric Technique. Numerical study have been done for Analysis of Mean number of low priority customers in the orbit (MNCO), Mean number of high priority customers in the queue(MPQL),Truncation level (OCUT),Probability of server free and Probabilities of server busy with low and high priority customers for various values of λ1 , λ2 , λ3 , μ1 , μ2 ,σ and k in elaborate manner and also various particular cases of this model have been discussed.

The planning and scheduling of operating rooms: A simulation approach

This paper considers the stochastic off-line planning and on-line scheduling of operating rooms (OR) at the operational level. It studies the problem for a local public hospital that adopts a closed block scheduling strategy for elective surgical cases. Emergency cases have their dedicated ORs; thus are not considered. At the off-line level, the coordinating surgeon of a given department assigns a subset of elective surgeries to each OR of the operating theater (OT). At the on-line level, unless mandated by extraneous factors, a surgical crew performs all surgical cases assigned to the OR, sequencing them in a non-increasing order of their largest expected surgical time.This paper investigates how to enhance the OT’s expected under and over utilization while maintaining the current average number of surgical cases in general and of major ones in particular. Specifically, it considers two methods that reduce the over utilization of the OT: canceling surgical cases at the on-line operational level, and limiting the workload planned at the off-line operational level. In addition, it considers three management strategies that monitor and control the flow of surgical cases with the objective of diminishing the variability of OR completion times. These strategies are: transferring a surgical case from a busy OR to a free one, using a single queue for all ORs, and adopting an alternative set up of the OT where surgical cases are separated by type.The paper builds a simulation model for each strategy, and compares the model’s output to that of the current situation. It assesses the utility of each strategy based on statistical inferential techniques. Regarding the OR over and under utilization, it is recommended to cancel surgical cases that start after the closing time of the OT. Since this may be difficult to implement, it is advisable to reduce the workload planned at the off-line operational level to 90% of the OT’s capacity. Regarding the variability of the OR completion times, it is judicious to adopt a single queue for all surgical cases. When this is infeasible from a technical or a managerial point of view, mixing surgical cases is preferable to separating them by type unless the hospital further decreases the number of minor cases assigned to the OT. Finally, the transfer of the last surgical case from a busy OR to a free one reduces the range of OR completion times.The proposed simulation model can be easily extended to other hospitals and/or to the strategic and tactical managerial levels. It can account for different constraints and/or managerial procedures. It can be used in conjunction with optimization techniques. Its implementation requires limited knowledge of basic simulation techniques while it offers a simple, user friendly, interactive, decision support system that can be used by coordinating surgeons and management.

STABILIZING PERFORMANCE IN NETWORKS OF QUEUES WITH TIME-VARYING ARRIVAL RATES

This paper investigates extensions to feed-forward queueing networks of an algorithm to set staffing levels (the number of servers) to stabilize performance % at Quality of Service (QoS) targets in anMt/GI/st+GImulti-server queue with a time-varying arrival rate. The model has a non-homogeneous Poisson process (NHPP), customer abandonment, and non-exponential service and patience distributions. For a single queue, simulation experiments showed that the algorithm successfully stabilizes abandonment probabilities and expected delays over a wide range of Quality-of-Service (QoS) targets. A limit theorem showed that stable performance at fixed QoS targets is achieved asymptotically as the scale increases (by letting the arrival rate grow while holding the service and patience distributions fixed). Here we extend that limit theorem to a feed-forward queueing network. However, these fixed QoS targets provide low QoS as the scale increases. Hence, these limits primarily support the algorithm with a low QoS target. For a high QoS target, effectiveness depends on the NHPP property, but the departure process never is exactly an NHPP. Thus, we investigate when a departure process can be regarded as approximately an NHPP. We show that index of dispersion for counts is effective for determining when a departure process is approximately an NHPP in this setting. In the important common case when all queues have high QoS targets, we show that both: (i) the departure process is approximately an NHPP from this perspective and (ii) the algorithm is effective.

Adaptive Frequency Control of Packet Processing Engines in Telecommunication Networks

This letter addresses the problem of energy saving in packet processing engines of network devices. The case of a single traffic queue is considered. The aim is to guarantee a delay constraint while adjusting the power consumption and the queue service rate. A feedback control is used, which is adaptive to the statistical behavior of traffic sources. The feedback is exploited by a heuristics and an optimal control technique. The latter approach, in particular, is capable of driving indications about which system conditions (statistical parameters and buffer size) are potentially more sensitive to energy saving.

A two-stage approach in solving the state probabilities of the multi-queue M/G/1 model

The M/G/1 model is the fundamental basis of the queueing system in many network systems. Usually, the study of the M/G/1 is limited by the assumption of single queue and infinite capacity. In practice, however, these postulations may not be valid, particularly when dealing with many real-world problems. In this paper, a two-stage state-space approach is devoted to solving the state probabilities for the multi-queue finite-capacity M/G/1 model, i.e. q-M/G/1/Ki with Ki buffers in the ith queue. The state probabilities at departure instants are determined by solving a set of state transition equations. Afterward, an embedded Markov chain analysis is applied to derive the state probabilities with another set of state balance equations at arbitrary time instants. The closed forms of the state probabilities are also presented with theorems for reference. Applications of Little's theorem further present the corresponding results for queue lengths and average waiting times. Simulation experiments have demonstrated the correctness of the proposed approaches.

Performance analysis of single server non-Markovian retrial queue with working vacation and constant retrial policy

This paper analyses an M/G/1 retrial queue with working vacation and constant retrial policy. As soon as the system becomes empty, the server begins a working vacation. The server works with different service rates rather than completely stopping service during a vacation. We construct the mathematical model and derive the steady-state queue distribution of number of customer in the retrial group. The effects of various performance measures are derived.

NUMBER OF RETRIALS IN A FINITE SOURCE RETRIAL QUEUE WITH UNRELIABLE SERVER

The object of this paper is to continue investigation of a single server retrial queue with finite number of sources in which the server is subjected to breakdowns and repairs. The server life time as well as the intervals between repetitions are exponentially distributed, while the repair and the service times are generally distributed. Using the formulas for the stationary system state distributions, obtained by Wang et al. [in Wang, J, L Zhao and F Zhang (2011). Analysis of the finite source retrial queues with server breakdowns and repairs. Journal of Industrial and Management Optimization, 7, 655–676.] we investigate the distribution of the number of retrials, made by a customer before he reaches the server free. Recurrent schemes for computing this distribution in steady state as well as any arbitrary of its moments are established. Numerical results for five different distributions of the service and repair times are also presented.

Paying for Express Checkout: Competition and Price Discrimination in Multi-Server Queuing Systems

We model competition between two firms selling identical goods to customers who arrive in the market stochastically. Shoppers choose where to purchase based upon both price and the time cost associated with waiting for service. One seller provides two separate queues, each with its own server, while the other seller has a single queue and server. We explore the market impact of the multi-server seller engaging in waiting cost-based-price discrimination by charging a premium for express checkout. Specifically, we analyze this situation computationally and through the use of controlled laboratory experiments. We find that this form of price discrimination is harmful to sellers and beneficial to consumers. When the two-queue seller offers express checkout for impatient customers, the single queue seller focuses on the patient shoppers thereby driving down prices and profits while increasing consumer surplus.