Expected Queue Length Research Articles

Modelling and analysis of hub-and-spoke networks under stochastic demand and congestion

Motivated by the strategic importance of congestion management, in this paper we present a model to design hub-and-spoke networks under stochastic demand and congestion. The proposed model determines the location and capacity of the hub nodes and allocate non-hub nodes to these hubs while minimizing the sum of the fixed cost, transportation cost and the congestion cost. In our approach, hubs are modelled as spatially distributed M/G/1 queues and congestion is captured using the expected queue lengths at hub facilities. A simple transformation and a piecewise linear approximation technique are used to linearize the resulting nonlinear model. We present two solution approaches: an exact method that uses a cutting plane approach and a novel genetic algorithm based heuristic. The numerical experiments are conducted using CAB and TR datasets. Analysing the results obtained from a number of problem instances, we illustrate the impact of congestion cost on the network topology and show that substantial reduction in congestion can be achieved with a small increase in total cost if congestion at hub facilities is considered at the design stage. The computational results further confirm the stability and efficiency of both exact and heuristic approaches.

Unreliable two-server queueing system with variable repair rates

This article considers an infinite-capacity two-server queue system, where the servers are subject to failure at any time. A repairman is responsible for repair the failure server and the repair rate is changeable depending on the number of failure servers. The repair rate of only one server is failure and that of two servers are all failure may be different. The steady-state reliability indexes and the stability condition of queue are derived through the transfer rate matrix. The expected queue length and some related steady-state performances are obtained by the probability-generating function transform method. Numerical examples are provided for illustration purpose.

Routing Heterogeneous Jobs to Heterogeneous Servers: A Global Optimization-Based Approach

We study the problem of routing heterogeneous jobs to heterogeneous servers, where the service rate depends both on the job type and the server type. We employ a math programming approach which is desirable since it can easily be embedded within larger planning problems. While convex objectives which minimize expected waiting time or expected sojourn time are well-studied in this context, we study a particular class of nonconvex objective functions that can be factored by arrival rate and workload. These include maximizing the service level, i.e., the probability that a job is served within an acceptable waiting time, and minimizing the expected queue length. We develop an optimization algorithm called fixed-ratio shifting envelopes to find near-optimal solutions to our math program, and design online routing policies which make use of these solutions. To showcase our model, we apply our model and solution method to the problem of assigning Irvine census block groups to fire stations.

Modelling and analysis of a bulk service queueing model with multiple working vacations and server breakdown

In this paper, a single server queue with variable batch size service, Poisson bulk arrival with multiple working vacations and server breakdown is considered. In working vacation, the server works with different rates rather than completely stoping the service during the vacation period. In this model, during the working vacation the server starts the service if it finds at least one customer in the queue with a maximum of ‘b’ customers, otherwise the server serves with variable batch size. Service time in working vacation and in regular period follows general distribution. The probability generating function of a queue size at an arbitrary time epoch as well as other completion epochs is derived. Expected queue length in a steady state is obtained. Also a numerical illustration is presented.

Steady state analysis of bulk arrival and bulk service queueing model with multiple working vacations

This paper studies a single server queue with variable batch size general service, Poisson bulk arrival and multiple working vacations where the server works with different rate rather than completely stops the service during the vacation period. In this model, the servers with variable batch size (a, b) in regular service period whereas, during the working vacation period the server starts the service if he finds even one customer in queue with a maximum of 'b' customers with different service time. Service time in working vacation and in regular period follows general distribution. The probability generating function of a queue size at an arbitrary time epochs as well as other completion epochs are derived. Expected queue length in a steady state is also obtained. Numerical illustration is presented.

Reward processes and performance simulation in supermarket models with different servers

Supermarket models with different servers have become a key in modelling resource management of stochastic networks, such as computer networks, manufacturing systems, transportation networks, and healthcare systems. However, the different servers always make analysis of such a supermarket model more interesting, difficult and challenging. This paper provides a novel method for analysing the supermarket models with different servers through a multi-dimensional continuous-time Markov reward process. Firstly, some utility functions are constructed for designing the routine selection mechanism according to the queue lengths, the service rates, and the probability of individual preference. Secondly, using the state jump points of the continuous-time Markov reward process, some segmented stochastic integrals of the random reward function are established by means of an event-driven technique. Based on this, the mean of the random reward function in a finite time interval is computed, and the mean of the discounted random reward function in an infinite time interval can also be calculated. Finally, some simulation experiments are given to indicate how the expected queue length of each server depends on some key parameters of this supermarket model.

Robust queueing theory: an initial study using imprecise probabilities

We study the robustness of performance predictions of discrete-time finite-capacity queues by applying the framework of imprecise probabilities. More concretely, we consider the Geo/Geo/1/L model with probabilities of arrival and departure that are no longer fixed, but are allowed to vary within given intervals. We distinguish between two concepts of independence in this framework, namely repetition independence and epistemic irrelevance. In the first approach, we assume the existence of time-homogeneous probabilities for arrival and departure, which leads us to consider a collection of stationary queues. In the second, the stationarity assumption is dropped and we allow the arrival and departure probabilities to vary from time point to time point; they may even depend on the complete history of queue lengths. We calculate bounds on the expected queue length, the probability of a particular queue length and the probability of turning on the server. For the expected queue length, both approaches coincide. For the other performance measures, we observe and discuss various differences between the bounds obtained for these two approaches. One of our observations is that ergodicity may break down due to imprecision: bounds on expected time averages of certain functions on the state space are not necessarily equal to the bounds on the expectation of that function at random instants in a steady-state queue.

ONE-DIMENSIONAL STOCHASTIC MODELS WITH OPEN BOUNDARIES: INTEGRABILITY, APPLICATIONS AND -DEFORMED KNIZHNIK–ZAMOLODCHIKOV EQUATIONS

This thesis contains work on three separate topics, but with common themes running throughout. These themes are drawn together in the asymmetric exclusion process (ASEP) – a stochastic process describing particles hopping on a one-dimensional lattice. The open boundary ASEP is set on a finite length lattice, with particles entering and exiting at both boundaries. The transition matrix of the open boundary ASEP provides a representation of the two boundary Temperley–Lieb algebra, and the integrability of the system allows the diagonalisation of the transition matrix through the Bethe ansatz method. We study the ASEP in the reverse bias regime, where the boundary injection and extraction rates oppose the preferred direction of flow in the bulk. We find the exact asymptotic relaxation rate along the coexistence line by analysing solutions of the Bethe equations. The Bethe equations are first solved numerically, then the form of the resulting root distribution is used as the basis for an asymptotic analysis. The reverse bias induces the appearance of isolated roots, which introduces a modified length scale in the system. We describe the careful treatment of the isolated roots that is required in both the numerical procedure, and in the asymptotic analysis. The second topic of this thesis is the study of a priority queueing system, modelled as an exclusion process with moving boundaries. We call this model the prioritising exclusion process (PEP). In the PEP, the hopping of particles corresponds to high priority customers overtaking low priority customers in order to gain service sooner. Although the PEP is not integrable, techniques from the ASEP allow calculation of exact density profiles in certain phases, and the calculation of approximate average waiting times when the expected queue length is finite. The final topic of this thesis is a study of polynomial solutions of a q-deformed Knizhnik–Zamolodchikov (qKZ) equation with mixed boundaries. The qKZ equation studied here is given in terms of the one boundary Temperley–Lieb algebra, and its solutions have a factorised form in terms of Baxterized elements of the type B Hecke algebra. We find an integral form for certain components of the qKZ solution, along with

Optimal Control of an N-Policy Two-Phase MX/Ek/1 Queuing System with Server Startup Subject to the Server Breakdowns and Delayed Repair

Background/Objective: To investigate the control policy of the two-phase Mx/Ek/1 queueing system with server startup time, N-policy and unreliable server, consisting of breakdown and delay periods. Methods: Steady state equations governing the queuing system are written and is solved using the probability generating functions to obtain the steady state probabilities when the server is in different states. Results: Expected queue length when the server is in different states is derived. Also the expected waiting time in the queue is obtained through heuristic approach. Optimal threshold of N that minimizes the average cost per unit time is derived. Sensitivity analysis is performed on the optimal threshold N* based on changes in the system parameters and cost elements for the geometric arrival batch size distribution. Conclusions/Applications: This model can be obtained in the two-phase service systems like communication networks and production systems to analyze the system performance.

On human mobility assisted data collection in wireless sensor networks

Mobile element based data collection has been developed for wireless sensor networks. In the existing work, mobility controlled devices are usually needed. We propose a data collection scheme based on human mobility in this paper, which is more energy efficient. We analyse its performance theoretically. Our work is based on the SLAW mobility model, which has been proposed recently according to the features of human mobility. We develop mathematical methods to calculate the expected waiting time for the sensing data and the expected queue length at the sensors. The correctness of our methods has been proved through numerical simulations.

Analysis of a clearing queueing system with setup times

We consider a Markovian clearing queueing system with setup times. When the system is empty, the server gets into the state of vacation. Once a new customer arrives the system, an exponential setup time is required before the server renders the service again. The customers are accumulated according to Poisson arrival process and the service times are exponentially distributed. Upon their arrivals, customers decide whether to join or balk the queue based on a natural linear reward-cost structure which reflects their desire for service and their unwillingness to wait. According to the state of server under some condition, we obtain the balking strategies of customers, the stationary distribution of system state, the expected queue length and the social optimal benefit. Finally, some numerical experiments describe how the expected queue length and the social optimal benefit depend on the arrival rate, the service time and the setup time.

Study of Queuing System of a Busy Restaurant and a Proposed Facilitate Queuing System

In a restaurant waiting for service is a common phenomenon for the customer. Restaurant owners are not concerned about the matter of waiting customer though they do not want their customer would go another competitor's door. There are several determining factors for a restaurant to be considered as a good or a bad one. Taste, cleanliness, the restaurant layout and settings are some of the most important factors. These factors, when managed carefully, will be able to attract plenty of customers, However, there is also another factor that needs to be considered especially when the restaurant has already succeeded in attracting customers. This factor is the customers queuing time. Queuing theory is the study of queue or waiting lines. Some of the analysis that can be derived using queuing theory include the expected waiting time in the queue, the average time in the system, the expected queue length, the expected number of customers served at one time, the probability of balking customers, as well as the probability of the system to be in certain states, such as empty or full. Hence, queuing theory is suitable to be applied in a restaurant setting since it has an associated queue or waiting line where customers who cannot be served immediately have to queue (wait) for service. Researchers have previously used queuing theory to model the restaurant operation reduce cycle time in a busy fast food restaurant as well as to increase throughput and efficiency. In this study, the average service time, average idle time, and average waiting time at cash counter are measured.

M/M/R+r machining system with reneging, spares and interdependent controlled rates

The present investigation deals with the multi component machine repair problem by including the concepts of reneging and interdependent controlled rates. We consider a machining system consisting of M operating units, S 1 and S 2 warm spares of two kinds and a repair facility having R permanent repairmen and r additional removable repairmen. When all the spares are used and the operating units fail, the system works in degraded mode. By using interdependent controlled arrival and service rates and reneging parameters along with time-sharing concept, the queueing model is developed for more versatile situations encountered in production/ manufacturing industries. The solution is obtained by using recursive technique. The expressions for the queue size distribution and expected queue length are established. Some more performance indices such as expected number of failed units waiting for the service, expected number of busy repairmen, expected number of both types of spare parts functioning as standbys, etc., are obtained with the help of recursive technique. By taking numerical illustrations, the sensitivity analysis is provided to validate the analytical results.

A Reliable Transport Protocol with Prediction Mechanism for Urgent Information in Wireless Sensor Networks

A reliable transport protocol with prediction mechanism for urgent information (PMUI) in WSNs is proposed. With PMUI, which is based on RTP-UI, the congestion control mechanism is improved, and the priority control and prediction mechanisms are adopted by taking the current queue length with change rate and expected queue length with remaining length together into consideration. The congestion status of current queue is analyzed and the changing trend of the next cycle of the queue is predicted as well. In order to evaluate the degree of congestion, state machine is adopted. Working states of WSNs are classified into eight states in accordance with different degrees of congestion. According to the change rate of a queue and the expected change rate of a queue, different rate adjustment mechanisms and bandwidth allocation schemes based on different working states are developed. Simulation results show that PMUI is lower than RTP-UI in both packet loss rate and the average delay for reliable transmission of urgent information.

Approximate queue length distribution of a discriminatory processor sharing queue with impatient customers

We consider a two-class processor sharing queueing system with impatient customers. The system operates under the discriminatory processor sharing (DPS) scheduling. The arrival process of each class customers is the Poisson process and the service requirement of a customer is exponentially distributed. The reneging rate of a customer is a constant. To analyze the performance of the system, we develop a time scale decomposition approach to approximate the joint queue-length distribution of each class customers. Via a numerical experiment, we show that the time scale decomposition approach gives a fairly good approximation of the queue-length distribution and the expected queue length.

Resource Allocation Using Queuing Theory In A Walk-In Clinic

The objective of this research is to produce a simple tool to assist health care management in quantifying the tradeoffs between different resource allocations. In many cases, intuition results in an appropriate selection; the quandary, however, is typically over the magnitude of improvement. The problem addressed herein uses queuing theory in the context of a hypothetical walk-in clinic. Different resource allocations are compared on the basis of the expected number of patients in the waiting room. Without comparative numbers, managers are forced to guesstimate the difference in expected queue lengths. Fact-based decisions not only improve quality but give the decision maker a sense of comfort.

Remarks on queue inference from departure data alone and the importance of the queue inference engine

Recently a generalization of Larson’s Queue Inference Engine (Q.I.E.) for computing conditional expected queue length was proposed in this journal. Although the conditional expectation calculation is correct and of independent interest, it does not represent the queuing inference quantities claimed. We demonstrate this with two examples with identical mean service times where the expectations claimed differ. If customer departure times are included, the examples demonstrate that queue inference cannot be made from departure data alone with unknown service distribution.

Asymptotically tight steady-state queue length bounds implied by drift conditions

The Foster–Lyapunov theorem and its variants serve as the primary tools for studying the stability of queueing systems. In addition, it is well known that setting the drift of the Lyapunov function equal to zero in steady state provides bounds on the expected queue lengths. However, such bounds are often very loose due to the fact that they fail to capture resource pooling effects. The main contribution of this paper is to show that the approach of “setting the drift of a Lyapunov function equal to zero” can be used to obtain bounds on the steady-state queue lengths which are tight in the heavy-traffic limit. The key is to establish an appropriate notion of state-space collapse in terms of steady-state moments of weighted queue length differences and use this state-space collapse result when setting the Lyapunov drift equal to zero. As an application of the methodology, we prove the steady-state equivalent of the heavy-traffic optimality result of Stolyar for wireless networks operating under the MaxWeight scheduling policy.

Stability and delay of distributed scheduling algorithms for networks of conflicting queues

This paper explains recent results on distributed algorithms for networks of conflicting queues. At any given time, only specific subsets of queues can be served simultaneously. The challenge is to select the subsets in a distributed way to stabilize the queues whenever the arrival rates are feasible. One key idea is to formulate the subset selection as an optimization problem where the objective function includes the entropy of the distribution of the selected subsets. The dual algorithm for solving this optimization problem provides a distributed scheduling algorithm that requires only local queue-length information. The algorithm is based on the CSMA (Carrier Sense Multiple Access) protocol in wireless networks. We also explain recent results, some of them unpublished so far, on the delay properties of these algorithms. In particular, we present a framework for queuing stability under bounded CSMA parameters, and show how the expected queue lengths depend on the throughput region to be supported. When the arrival rates are within a fraction of the capacity region, queue lengths that are polynomial (or even logarithmic) in the number of queues can be achieved.

Synchronous working vacation policy for finite-buffer multiserver queueing system

This paper presents modeling and analysis of unreliable Markovian multiserver finite-buffer queue with discouragement and synchronous working vacation policy. According to this policy, c servers keep serving the customers until the number of idle servers reaches the threshold level d; then d idle servers take vacation altogether. Out of these d vacationing servers, d W servers may opt for working vacation i.e. they serve the secondary customers with different rates during the vacation period. On the other hand, the remaining d − d W = d V servers continue to be on vacation. During the vacation of d servers, the other e = c − d servers must be present in the system even if they are idle. On returning from vacation, if the queue size does not exceed e, then these d servers take another vacation together; otherwise start serving the customers. The servers may undergo breakdown simultaneously both in regular busy period and working vacation period due to the failure of a main control unit. This main unit is then repaired by the repairman in at most two phases. We obtain the stationary performance measures such as expected queue length, average balking and reneging rate, throughput, etc. The steady state and transient behaviours of the arriving customers and the servers are examined by using matrix analytical method and numerical approach based on Runge–Kutta method of fourth order, respectively. The sensitivity analysis is facilitated for the transient model to demonstrate the validity of the analytical results and to examine the effect of different parameters on various performance indices.