General Service Time Research Articles

Waiting Time Control Chart for M/G/1 Retrial Queue

Retrial queues are used extensively to model many practical problems in service systems, call centers, data centers, and computer network systems. The average waiting time is the main observable characteristic of the retrial queues. Long queues may cause negative impacts such as waste of manpower and unnecessary crowding leading to suffocation, and can even cause trouble for customers and institutions. Applying control chart technology can help managers analyze customers’ waiting times to improve the effective performance of service and attention. This paper pioneers the developing and detailed study of a waiting time control chart for a retrial queue with general service times. Two waiting time control charts, the Shewhart control chart, and a control chart using the weighted variance method are constructed in this paper. We present three cases for the Shewhart control chart in which the service time obeys special distributions, such as exponential, Erlang, and hyper-exponential distributions. The case of an exponentially distributed service time is also presented for the control chart using the weighted variance method. Based on the numerical simulations conducted herein, managers can better monitor and analyze the customers’ waiting times for their service systems and take preventive measures.

Performance Modeling and Analysis of P4 Programmable Devices With General Service Times

Performance Modeling and Analysis of P4 Programmable Devices With General Service Times

Regenerative Analysis and Approximation of Queueing Systems with Superposed Input Processes

A single-server queueing system with n classes of customers, stationary superposed input processes, and general class-dependent service times is considered. An exponential splitting is proposed to construct classical regeneration in this (originally non-regenerative) system, provided that the component processes have heavy-tailed interarrival times. In particular, we focus on input processes with Pareto interarrival times. Moreover, an approximating GI/G/1-type system is considered, in which the independent identically distributed interarrival times follow the stationary Palm distribution corresponding to the stationary superposed input process. Finally, Monte Carlo and regenerative simulation techniques are applied to estimate and compare the stationary waiting time of a customer in the original and in the approximating systems, as well as to derive additional information on the regeneration cycles’ structure.

Level-Crossing Characteristics of an Actively Managed Buffer

In this paper, we examine a buffer with active management that rejects packets basing on the buffer occupancy. Specifically, we derive several metrics characterizing how effectively the algorithm can prevent the queue of packets from becoming too long and how well it assists in flushing the buffer quickly when necessary. First, we compute the probability that the size of the queue is kept below a predefined level L. Second, we calculate the distribution of the amount of time needed to cross level L, the buffer overflow probability, and the average time to buffer overflow. Third, we derive the distribution of the amount of time required to flush the buffer and its average value. A general modeling framework is used in derivations, with a general service time distribution, general rejection function, and a powerful model of the arrival process. The obtained formulas enable, among other things, the solving of many design problems, e.g., those connected with the design of wireless sensor nodes using the N-policy. Several numerical results are provided, including examples of design problems and other calculations.

Orbit while in service

In various real-life queueing systems, part of the service can be rendered without involvement or presence of the customers themselves. In those queues, customers whose service order is still in process may leave the service station, go to ‘orbit’ for a random length of time, and then return to find out if their order has been completed. Common examples are car’s annual maintenance works, food ordering, etc. In this paper, a thorough analysis of a single-server ‘orbit while in service’ queueing model with general service time is presented. Assuming an Exponentially distributed orbit time, we derive general formulae for the distributions of (i) a customer’s total residence time in the system; (ii) a customer’s net actual residence time in the system during service (not including orbit time); (iii) the time an orbiting customer is late to return, i.e., remains in orbit after his/her service has been completed; and (iv) the total number of customers in the system. Considering the family of Gamma-distributed service times (spanning the range of distributions between the Exponential and the Deterministic), as well as the Uniform distribution, we further derive explicit formulae for the distributions of the above variables. Under linear cost assumptions, the optimal mean orbit time is numerically calculated for each of the above service-time distributions. Figures depicting the behavior of the measures as functions of the parameters are presented.

Efficient scheduling in redundancy systems with general service times

Efficient scheduling in redundancy systems with general service times

Non Poisson Queue Data Analysis at KFC Jimbaran Regular Service

In providing services, it is not uncommon to be faced with a high level of customer arrival so that they have to wait their turn to get service. This waiting line is called a queue, one of which is non- poisson queue. An activity includes a non-poisson queue when the distribution of arrivals or services is non-poisson or exponentially distributed. This research focused on examining the cashier service queue at Kentucky Fried Chicken Fast Food Restaurant in Jimbaran, Bali. Data collection by direct observation was carried out for 4 hours 2 minutes. Service facilities in the system under study are 2 waiters. Researchers made observations of arrival time, queuing time data, service time, the number of queuing customers, and the number of customers served. Next, this data is processed and produces data on the number of Poisson-distributed arrivals, Weibull-distributed inter-arrival times, and General Pareto-distributed service times. Through this, it is concluded that the queuing model at KFC is (M/G/2): (FIFO/∞/∞). After calculating to determine the system performance measure in this model, a utilization factor of 50.75% is obtained. The percentage of the utilization factor indicates that the KFC service facility is not too busy serving customers, so there is no need to add service facilities in the system.

Open networks of infinite server queues with non-homogeneous multivariate batch Poisson arrivals

In this paper, we consider the occupancy distribution for an open network of infinite server queues with multivariate batch arrivals following a non-homogeneous Poisson process, and general service time distributions. We derive a probability generating function for the transient occupancy distribution of the network and prove that it is necessary and sufficient for ergodicity that the expected occupancy time for each batch be finite. Further, we recover recurrence relations for the transient probability mass function formulated in terms of a distribution obtained by compounding the batch size with a multinomial distribution.

Post-processing Improvements in Multi-objective Optimization of General Single-server Finite Queueing Networks

An alternative mathematical programming formulation is considered for a mixed-integer optimization problem in queueing networks. The sum of the blocking probabilities of a general service time, single server, and the finite, acyclic queueing network is minimized, and so are the total buffer sizes and the overall service rates. A multi-objective genetic algorithm (MOGA) and a particle swarm optimization (MOPSO) algorithm are combined to solve this difficult stochastic problem. The derived algorithm produces a set of efficient solutions for multiple objectives in the objective function. The implementation of the optimization algorithms is dependent on the generalized expansion method (GEM), a classical tool used to evaluate the performance of finite queueing networks. We carried out a set of computational experiments to attest to the efficacy and efficiency of the proposed approach. In addition, we present a comparative analysis of the solutions before and after post-processing. Insights obtained from the study of complex queue networks may assist the planning of these types of queueing networks.

Stability analysis of two-class retrial systems with constant retrial rates and general service times

We establish stability criterion for a two-class retrial system with Poisson inputs, general class-dependent service times and class-dependent constant retrial rates. We also characterize an interesting phenomenon of partial stability when one orbit is tight but the other orbit goes to infinity in probability. All theoretical results are illustrated by numerical experiments.

Performance analysis of flexible flow shop with parallel manufacturing cells

Performance evaluation for stochastic manufacturing systems is significant for system design optimization, but it is difficult to exactly analyse these systems with general arrival processes and service times. This paper studies a flexible flow shop with parallel manufacturing cells and models it as a queueing network. Parallel manufacturing cells and finite pallets lead to strong dependence among stages, and existing methods cannot be directly employed in this problem. Thus, approximate analysis is proposed to estimate performance measures of the flexible flow shop with parallel manufacturing cells based on two-moment approximations for a parallel-input synchronization station and an aggregation procedure. Numerical studies show the accuracy and the efficiency of the proposed method by comparing it with simulation results of experiments. This work can provide a basis to analyse and design more complex manufacturing systems.

Finite capacity single-server queue with Poisson input, general service and delayed renovation

This paper is devoted to the performance evaluation of the finite–capacity single-server queue with Poisson input and general service times, which is equipped with the novel queue-control mechanism — called renovation with delay. Renovation implies that each customer, when leaving the system upon its service completion, may also remove customers waiting in the queue. The number of customers to be removed is selected each time independently of the previous decisions and probabilistically based on the given probability distribution. The renovation with the delay considered in this paper means that the system must accumulate the (known) fixed number of service completions before the next renovation can be triggered. First in the literature numerical evidence is given, that the long-run performance of the system with such delayed renovation mechanism can be similar to that of the systems with the well-known active queue management mechanisms such as random early detection, at least in its basic version. The key stationary performance characteristics reported are the loss probability and the mean number of customers in the system. The unified analytical framework for the computation of the key performance characteristics, which is of independent interest, is also developed.

Per-flow structure of losses in a finite-buffer queue

We analyze the structure of losses in individual flows, in a multi-flow, finite-buffer queueing system. Namely, a model with many separate flows (streams) of jobs arriving to a shared buffer, where they are subject to losses due to buffer overflows, is considered. (Such systems are common, for instance, in computer networking, where flows of packets arrive to the same router’s buffer from different network users). Assuming a general service time distribution and Poisson flows, we study the burst ratio parameter, which reflects the tendency of losses to cluster together, in long series. In particular, an explicit formula for the burst ratio in each individual flow is derived. Using this formula we show, among other things, that the per-flow burst ratio may vary significantly among flows and differ from the global burst ratio. This distinguishes the per-flow burst ratio from the per-flow loss ratio, which is the same for all flows. We demonstrate also the dependence of the per-flow burst ratio on the flow rate, number of flows, buffer size, system load and variance of the service time.

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.

Appointment Scheduling for Multiple Servers

Appointment schedules, in essence, balance supply and demand and are often employed in settings where resources are scarce and thus a high utilization is realized (e.g., healthcare). Whereas most of the existing literature focuses on the single-server case, a framework is developed to study appointment scheduling in multiserver settings. Relying on phase-type approximations, general service-time distributions are modeled, which are fed into a recursive approach allowing evaluation and optimization of an objective function that balances expected waiting times and idle times. Studying optimized schedules for multiple servers reveals that the start and end of a session can deviate greatly from the dome-shaped pattern as established for the single-server case. Furthermore, a comparison of various multiserver setups shows that significant performance gains can be achieved when servers are pooled. This allows an explicit quantification of the cost of continuity of care. In addition, session overtime as well as early finish of servers can be incorporated in the approach; the benefits of the additional flexibility that a multiserver setting provides are summarized. For the stationary plateau of the dome, to which the optimal interarrival times converge, steady-state appointment schedules are obtained by exploiting the embedded Markov chain; these schedules are shown and argued to converge quickly to optimal solutions obtained in a heavy-traffic regime. In this regime, algebraic solutions are derived, which provide interesting managerial guidelines when the pooling of servers is considered in appointment scheduling. This paper was accepted by Bariş Ata, stochastic models and simulation.

Stochastic Timed Discrete-Event Systems: Modular Modeling and Performance Evaluation Through Markovian Jumps

We are interested in a scalable, flexible, and modular methodology, for modeling and performance analysis of stochastic discrete-event systems (SDES). In this sense, we propose a modular approach for timing non-markovian SDES expressed as a parallel composition of modules that interacts with each other through events. We show how general distribution for event lifetimes can be implemented systematically by coupling timing modules to the system model. As a result, this coupling mechanism preserves modularity, leading to a compact markovian model expressed in terms of flexible modules. Therefore the methodology allows us to write the whole SDES model as a composition of the system model and the timing one, giving flexibility and scalability in modeling design, as we can modify the modules individually according to the designer’s interests. In addition, from the whole markovian SDES model, we show how to perform the model analysis through the analytic approach, as well as through Monte Carlo computer simulation. As an application, we present a numerical example of computing the abandonment rate for a service network with general service time employing both analytical and computer-simulation models.

Analysis of a Discrete-Time Queueing Model with Disasters

Queueing models with disasters can be used to evaluate the impact of a breakdown or a system reset in a service facility. In this paper, we consider a discrete-time single-server queueing system with general independent arrivals and general independent service times and we study the effect of the occurrence of disasters on the queueing behavior. Disasters occur independently from time slot to time slot according to a Bernoulli process and result in the simultaneous removal of all customers from the queueing system. General probability distributions are allowed for both the number of customer arrivals during a slot and the length of the service time of a customer (expressed in slots). Using a two-dimensional Markovian state description of the system, we obtain expressions for the probability, generating functions, the mean values, variances and tail probabilities of both the system content and the sojourn time of an arbitrary customer under a first-come-first-served policy. The customer loss probability due to a disaster occurrence is derived as well. Some numerical illustrations are given.

On A Single Server Queue with Two-Stage First Essential Service Followed by One of the Two Types of Additional Optional Service and Optional Deterministic Server Vacations

We study the steady state behavior of a batch arrival single server queue in which the first service consisting of two stages with general service times G1 and G2 is compulsory. After completion of the two stages of the first essential service, a customer has the option of choosing one of the two types of additional service with respective general service times G1 and G2 . Just after completing both stages of first essential service with or without one of the two types of additional optional service, the server has the choice of taking an optional deterministic vacation of fixed (constant) length of time. We obtain steady state probability generating functions for the queue size for various states of the system at a random epoch of time in explicit and closed forms. The steady state results of some interesting special cases have been derived from the main results.

On the stability of queues with the dropping function.

In this paper, the stability of the queueing system with the dropping function is studied. In such system, every incoming job may be dropped randomly, with the probability being a function of the queue length. The main objective of the work is to find an easy to use condition, sufficient for the instability of the system, under assumption of Poisson arrivals and general service time distribution. Such condition is found and proven using a boundary for the dropping function and analysis of the embedded Markov chain. Applicability of the proven condition is demonstrated on several examples of dropping functions. Additionally, its correctness is confirmed using a discrete-event simulator.

Sampling and Remote Estimation for the Ornstein-Uhlenbeck Process Through Queues: Age of Information and Beyond

Recently, a connection between the age of information and remote estimation error was found in a sampling problem of Wiener processes: If the sampler has no knowledge of the signal being sampled, the optimal sampling strategy is to minimize the age of information; however, by exploiting causal knowledge of the signal values, it is possible to achieve a smaller estimation error. In this paper, we generalize the previous study by investigating a problem of sampling a stationary Gauss-Markov process named the Ornstein-Uhlenbeck (OU) process, where we aim to find useful insights for solving the problems of sampling more general signals. The optimal sampling problem is formulated as a constrained continuous-time Markov decision process (MDP) with an uncountable state space. We provide an exact solution to this MDP: The optimal sampling policy is a threshold policy on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">instantaneous estimation error</i> and the threshold is found. Further, if the sampler has no knowledge of the OU process, the optimal sampling problem reduces to an MDP for minimizing a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear</i> age of information metric. The age-optimal sampling policy is a threshold policy on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">expected estimation error</i> and the threshold is found. In both problems, the optimal sampling policies can be computed by low-complexity algorithms (e.g., bisection search and Newton’s method), and the curse of dimensionality is circumvented. These results hold for (i) general service time distributions of the queueing server and (ii) sampling problems both with and without a sampling rate constraint. Numerical results are provided to compare different sampling policies.