The analysis of a three-phase queueing servicing system with renewable servers

What determines the average length of a queue, which stretches in front of a service station? The answer to this question clearly depends on the average rate at which jobs arrive at the queue and on the average rate of service. Somewhat less obvious is the fact that stochastic fluctuations in service and arrival times are also important, and that these are a major source of backlogs and delays. Strategies that could mitigate fluctuations-induced delays are, thus in high demand as queue structures appear in various natural and man-made systems. Here, we demonstrate that a simple service resetting mechanism can reverse the deleterious effects of large fluctuations in service times, thus turning a marked drawback into a favorable advantage. This happens when stochastic fluctuations are intrinsic to the server, and we show that service resetting can then dramatically cut down average queue lengths and waiting times. Remarkably, this strategy is also useful in extreme situations where the variance, and possibly even mean, of the service time diverge—as resetting can then prevent queues from “blowing up.” We illustrate these results on the M/G/1 queue in which service times are general and arrivals are assumed to be Markovian. However, the main results and conclusions coming from our analysis are not specific to this particular model system. Thus, the results presented herein can be carried over to other queueing systems: in telecommunications, via computing, and all the way to molecular queues that emerge in enzymatic and metabolic cycles of living organisms.

Discrete-Time Approximation of the Machine Interference Problem with Generally Distributed Failure, Repair, and Walking Times

This paper presents a general framework for optimal task reallocation in heterogeneous distributed-computing systems and offers a rigorous analytical model for the stochastic execution time of a workload. The model takes into account the heterogeneity and stochastic nature of the tasks' service and transfer times, servers' failure times, as well as an arbitrary task-reallocation policy. The stochastic service, transfer and failure times are assumed to have general, age-dependent (non-exponential) distributions, resulting in a tandem distributed queuing system with non-Markovian dynamics. Auxiliary age variables are introduced in the analysis to capture the memory associated with the non-Markovian stochastic times, thereby enabling a regenerative age-dependent analytical characterization of the statistics of the execution time of a workload. The model is utilized to devise task reallocation policies that optimize three metrics: the average execution time of a workload, the quality-of-service in executing a workload by a prescribed deadline and the reliability in executing a workload. Implications of the non-exponential event times on these metrics are also studied. Key results are verified experimentally on a distributed-computing testbed.

Algorithm for computing the queue length distribution at various time epochs in DMAP/G(1, a, b)/1/N queue with batch-size-dependent service time

Out-patient medical clinics in Sri Lanka often have long queues of patients waiting to see a doctor partly due to a shortage of doctors. To apply operational research techniques to optimize the functioning of the clinic, an initial step is to determine the distributional pattern of patient arrival times and the doctors' service times. The results showed that the arrival times of patients were time dependent. However, analysis of half-hourly data of inter-arrival times showed a negative exponential distribution. The doctor's service time varied from doctor to doctor and whether new or follow up patients were being seen. Each of the service times seemed to fit into an Erlangian distribution.

Survival data consisting of independent sets of correlated failure times may arise in many situations. For example, we may take repeated observations of the failure time of interest from each patient or observations of the failure time on siblings, or consider the failure times on littermates in toxicological experiments. Because the failure times taken on the same patient or related family members or from the same litter are likely correlated, use of the classical log-rank test in these situations can be quite misleading with respect to type I error. To avoid this concern, this paper develops two closed-form asymptotic summary tests, that account for the intraclass correlation between the failure times within patients or units. In fact, one of these two test includes the classical log-rank test as a special case when the intraclass correlation equals 0. Furthermore, to evaluate the finite-sample performance of the two tests developed here, this paper applies Monte Carlo simulation and notes that they can actually perform quite well in a variety of situations considered here.

In this paper, we investigate the performance of batch service queue model with second optional service, repairable breakdown and warm standby server. Both primary operating and warm standby servers provide First Essential Service (FES) and Second Optional Service (SOS) to customers, wherein FES is all arriving customers and only some of them may further request the SOS. The service times, failure times and repair times of both primary operating and warm standby server are assumed to follow exponential distributions. We use Runge-Kutta method to obtain the transient state probabilities and matrix-decomposition method to obtain the steady-state probabilities of the model. Also, a cost model is presented to determine the optimal service rates so that the expected cost is minimised. Finally, the effect of the model parameters on the system behaviour is demonstrated through numerical results and discussions.

When Service Times Depend on Customers’ Delays: A Relationship Between Two Models of Dependence Service times of customers often depend on the delay they experience in queue, as was recently demonstrated empirically in restaurants, call centers, and intensive care units. Two forms of dependence mechanisms in service systems with customer abandonment are studied in this paper: First, the service requirement of a customer may evolve while waiting in queue. Second, customers may arrive to the system with an exogenous service and patience time that are stochastically dependent. Because either dependence mechanism can have significant impacts on a system's performance, it should be identified and taken into consideration for performance evaluation and decision-making purposes. However, identifying the source of dependence from observed data is hard because both the service times and patience times are censored due to customer abandonment. Further, even if the dependence is known to be the latter exogenous one, there remains the difficult task of fitting a joint service-patience times distribution to the censored data. In “When Service Times Depend on Customers’ Delays: A Relationship Between Two Models of Dependence”, Wu, Bassamboo, and Perry provide a solution to address these statistical challenges.

Cost-benefit analysis of a two-unit warm standby reliability system with two types of repair facilities

This paper deals with the analysis of two classes of batch arrivals, one is high priority and other is low priority(retrial) customers with non-preemptive priority service, working vacations, negative arrival, emergency vacation for an unreliable server, which consists of a break-down by negative arrival and delay time to start repair. When there are no customers in the queue and in the orbit at the time of service completion for a positive customers, the server goes for a working vacation. The server works at a lower service rate during working vacation period. If there are no customers in the system at the end of vacation, the server becomes idle and ready for serving the new arrivals with probability p(single working vacation) or it remains on vacation with probability $$q=(1{-}p)$$ (multiple working vacations). Here we assume that customers arrive according to compound Poisson process in which high priority customers are assigned to class one and class two customers are of a low priority type. The priority customers that find the server busy are queued and then served in accordance with FCFS discipline. The arriving low priority customers on finding the server busy are queued in the orbit in accordance with FCFS constant retrial policy. The arrival of negative customer remove the customer being in service and makes the server breakdown and the server fail for a short interval of time. When the server fails it will send to repair. But the repair time do not start immediately, there is a delay time to start the repair process. We consider an emergency vacation which means that during the service time the server suddenly go for a vacation and the interrupted customer waits to getting the remaining service. The retrial time, service time, emergency vacation time, delay time and repair time are all follows general(arbitrary) distribution, working vacation time and negative arrival follows exponential distribution. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results are obtained explicitly. Also some performance measures of our model such as the system state probabilities, reliability indices, the average number of customer in the high priority queue and the low priority in the orbit and the average waiting time are derived. Numerical results are computed and graphical representations are presented.

We consider an optimal control problem for the hybrid model of a deterministic flow shop system, in which the jobs are processed in the order they arrive at the system. The problem is decomposed into a higher-level discrete-event system control problem of determining the optimal service times, and a set of lower-level classical control problems of determining the optimal control inputs for given service times. We focus on the higher-level problem which is nonconvex and nondifferentiable. The arrival times are known and the decision variables are the service times that are controllable within constraints. We present an equivalent convex optimization problem with linear constraints. Under some cost assumptions, we show that no waiting is observed on the optimal sample path. This property allows us to simplify the convex optimization problem by eliminating variables and constraints. We also prove, under an additional strict convexity assumption, the uniqueness of the optimal solution and propose two algorithms to decompose the simplified convex optimization problem into a set of smaller convex optimization problems. The effects of the simplification and the decomposition on the solution times are shown on an example problem.

This paper is a study of staff issues at the bank, ICBC China. Due to extra and imbalanced staff, the system was idle most of the time. By using the qualitative approach, M/M/s queuing model and discrete event simulation, the staff was sub-optimized. First, observation of the system was conducted and recorded using the existing operations system used by the bank. Arrival time of customers, service time, and the waiting time customers are willing to wait are all included in the data collection. Second, we analyzed the inter arrival time and service time by using the quantitative method and forecast the improvement in the system. We find out that the utilization of the system was very ineffective. Next, we suggested qualitative approach in the designing of queuing system of the bank. Finally, building and validation of a computer simulation model of a bank queuing system was conducted by using POM-QM software. A quantitative analysis and simulation was conducted and sub-optimized the system. The resulting design has a fewer number of service representatives, higher utilization and a waiting time within the limit of the customer's expected time.

Restaurant industry has become one of the most profitable industries in the world where incessant long waiting time may not only lead to customers’ dissatisfactions but also facilitate loosing of customers to other competitors. In this paper, in order to determine customers’ arrival patterns and service patterns which are critical factors in determining customers’ queue length and waiting time for a given restaurant, the food serving process employed at a named International Institute Restaurant (IIR), Ibadan, Nigeria, was used as a case study. Data were collected on customers’ number, customers’ inter arrival time and service time from Monday to Friday for a week. The data were analyzed statistically using the ARENA Input Analyzer to determine the arrival patterns and service patterns of customers within five working days of the week (Monday, Tuesday, Wednesday, Thursday and Friday). The results of the data analysis revealed that the arrival times of customers who patronized the IIR on Monday and Tuesday followed a Beta distribution. Furthermore, the arrival times of customers patronizing the IIR on Wednesday and Thursday followed a Weibull distribution while that of Friday assumed an Erlang distribution. Besides, the results of the data analysis revealed that the service times at IIR on Monday and Tuesday followed a Lognormal distribution. Beta, Lognormal and Weibull distributions were recorded in respect of service times characterizing the IIR on Wednesday, Thursday and Friday, respectively.

In this paper we study a broad class of semi-Markovian queues introduced by Sengupta. This class contains many classical queues such as the GI/M/1 queue, SM/MAP/1 queue and others, as well as queues with correlated inter-arrival and service times. Queues belonging to this class are characterized by a set of matrices of size m and Sengupta showed that its waiting time distribution can be represented as a phase-type distribution of order m. For the special case of the SM/MAP/1 queue without correlated service and inter-arrival times the queue length distribution was also shown to be phase-type of order m, but no derivation for the queue length was provided in the general case. This paper introduces an order m^2 phase-type representation (kappa, K) for the queue length distribution in the general case. Moreover, we prove that the order m^2 of the distribution cannot be further reduced in general. Examples for which the order is between m and m^2 are also identified. We derive these results in both discrete and continuous time and also discuss the numerical procedure to compute (kappa, K). Moreover, by combining a result of Sengupta and Ozawa, we provide a simple formula to compute the order m phase-type representation of the waiting time in a MAP/MAP/1 queue without correlated service and inter-arrival times, using the R matrix of a Quasi-Birth-Death Markov chain.

A matrix geometric representation for the queue length distribution of multitype semi-Markovian queues