General Service Time Distributions Research Articles

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.

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.

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.

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.

The Stability Analysis of a Double-X Queuing Network Occurring in the Banking Sector

We model a common teller–customer interaction occurring in the Ghanaian banking sector via a Double-X queuing network consisting of three single servers with infinite-capacity buffers. The servers are assumed to face independent general renewal of customers and independent identically distributed general service times, the inter-arrival and service time distributions being different for each server. Servers, when free, help serve customers waiting in the queues of other servers. By using the fluid limit approach, we find a sufficient stability condition for the system, which involves the arrival and service rates in the form of a set of inequalities. Finally, the model is validated using an illustrative example from a Ghanaian bank.

Performance Analysis of Packet Aggregation Mechanisms and Their Applications in Access (e.g., IoT, 4G/5G), Core, and Data Centre Networks.

The transmission of massive amounts of small packets generated by access networks through high-speed Internet core networks to other access networks or cloud computing data centres has introduced several challenges such as poor throughput, underutilisation of network resources, and higher energy consumption. Therefore, it is essential to develop strategies to deal with these challenges. One of them is to aggregate smaller packets into a larger payload packet, and these groups of aggregated packets will share the same header, hence increasing throughput, improved resource utilisation, and reduction in energy consumption. This paper presents a review of packet aggregation applications in access networks (e.g., IoT and 4G/5G mobile networks), optical core networks, and cloud computing data centre networks. Then we propose new analytical models based on diffusion approximation for the evaluation of the performance of packet aggregation mechanisms. We demonstrate the use of measured traffic from real networks to evaluate the performance of packet aggregation mechanisms analytically. The use of diffusion approximation allows us to consider time-dependent queueing models with general interarrival and service time distributions. Therefore these models are more general than others presented till now.

An Efficient Convolution Algorithm for the Non-Markovian Two-Node Cyclic Network

Consider a closed cyclic queueing model that consists of two nodes and a total of M customers. Each node buffer can accommodate all M customers. Node 1 has N ≤ M servers, each having an exponential service time with rate λ . The second node consists of a single server with a general service time distribution function B . . The well-known machine repair model with spares, where a set of identical machines, N , is served by a single repair person, is a key application of this model. This model has several other applications in performance evaluation, manufacturing, computer networks, and in reliability studies as it can be easily used to compute system availability. In this article, we give an efficient algorithm to derive an exact solution for the steady state system size probabilities. Our approach is based on developing an efficient polynomial convolution method to compute the transition probabilities of a birth process over node 2 service times and solving an imbedded Markov chain at node 2 service completion epochs. This is a significant improvement over the exponential algorithm given in an earlier paper. Numerical examples are given to demonstrate the performance of our method.

Switching diffusion approximations for optimal power management in parallel processing systems

In this paper, we investigate optimal power management in parallel processing systems composed of one queue and several identical processing stations. Power consumption is controlled by setting some of the stations into an inactive state, where they consume less power but are unable to provide service. This way, we are faced with the conflicting objective of minimizing power consumption while maintaining a desired quality of service. A distinguishing feature here, regarding most previous literature on this subject, is that we consider systems operating a policy that may turn the reserve machines on or off with setup times and under general inter-arrival or service time distributions, subject to some conditions. When these condition fail, we also provide a model with general inter-arrival times and exponentially distributed service times. To some extent, a controlled switching diffusion obtained in this paper via heavy traffic analysis and stochastic optimal control theory are the technical underpinning of the paper. We also propose a numerical approach to the solutions of the optimal control problems based on the Markov chain approximation method. Finally, we consider some numerical experiments that illustrate the efficiency of the proposed approach.

Strategic arrivals to a queue with service rate uncertainty

We study the problem of strategic choice of arrival time to a single-server queue with opening and closing times when there is uncertainty regarding service speed. A Poisson population of customers choose their arrival time with the goal of minimizing their expected waiting times and are served on a first-come first-served basis. There are two types of customers that differ in their beliefs regarding the service-time distribution. The inconsistent beliefs may arise from randomness in the server state along with noisy signals that customers observe. Customers are aware of the two types of populations with differing beliefs. We characterize the Nash equilibrium dynamics for exponentially distributed service times and show how they substantially differ from the model with homogeneous customers. We further provide an explicit solution for a fluid approximation of the game. For general service-time distributions we provide an algorithm for computing the equilibrium in a discrete-time setting. We find that in equilibrium customers with different beliefs arrive during different (and often disjoint) time intervals. Numerical analysis further shows that the mean waiting time increases with the coefficient of variation of the service time. Furthermore, we present a learning agent-based model (ABM) in which customers make joining decisions based solely on their signals and past experience. We numerically compare the long-term average outcome of the ABM with that of the equilibrium and find that the arrival distributions are quite close if we assume (for the equilibrium solution) that customers are fully rational and have knowledge of the system parameters, while they may greatly differ if customers have limited information or computing abilities.

Discrete-Time Queueing Model of Age of Information With Multiple Information Sources

Information freshness in IoT-based status update systems has recently been studied through the Age of Information (AoI) and Peak AoI (PAoI) performance metrics. In this article, we study a discrete-time server arising in multisource IoT systems, which accepts incoming information packets from multiple information sources so as to be forwarded to a remote monitor for status update purposes. Under the assumption of Bernoulli information packet arrivals and a common general discrete phase-type service time distribution across all the sources, we numerically obtain the exact per-source distributions of AoI and PAoI in matrix-geometric form for three different queueing disciplines: 1) nonpreemptive bufferless; 2) preemptive bufferless; and 3) nonpreemptive single buffer with replacement. The proposed numerical algorithm employs the theory of discrete-time Markov chains of quasi-birth-death type and is matrix analytical. Numerical examples are provided to validate the accuracy and effectiveness of the proposed queueing model. We also present a numerical example on the optimum choice of the Bernoulli parameters in a practical IoT system with two sources with diverse AoI requirements.

A lower bound on the stability region of redundancy-[formula omitted] with FIFO service discipline

Due to the complex dynamics of systems with redundancy and cancellations, existing results on the stability region of the Redundancy-d (R(d)) load balancing policy are scarce and are limited to very special cases with respect to the joint service time distribution of tasks. We provide a non-trivial, closed form lower bound on the stability region the R(d) for a general joint service time distribution. While simulation results indicate our bound is not tight, it provides an easy-to-check performance guarantee.

A queueing-inventory system with random order size policy and server vacations

In this paper, we consider a queueing-inventory system under continuous review with a random order size policy and lost sales. If the inventory is depleted after the service of a customer, a replenishment order is instantaneously triggered. The replenishment order size may be randomized according to a discrete probability distribution. Customers arrive in the system according to a Poisson process and require service from a server. The server takes multiple vacations once the inventory is depleted. The service time, the lead time, and the vacation time are all assumed to be distributed exponentially. We derive the stationary joint distribution of the queue length, the on-hand inventory level, and the status of the server in explicit product form. Furthermore, the conditional distributions of the on-hand inventory level when the server is off due to a vacation or depleted inventory, and when the server is on and working, are derived. Then, we calculate some of the system performance measures. The effect of the server’s vacation on the performance measures is investigated analytically. Finally, some numerical results are presented. The simulation study of the model in the context of more general arrival processes and service time distributions is presented.

On the Age of Information in Multi-Source Queueing Models

Freshness of status update packets is essential for enabling services where a destination needs the most recent measurements of various sensors. In this paper, we study the information freshness of single-server multi-source queueing models under a first-come first-served (FCFS) serving policy. In the considered model, each source independently generates status update packets according to a Poisson process. The information freshness of the status updates of each source is evaluated by the average age of information (AoI). We derive an exact expression for the average AoI for the case with exponentially distributed service time, i.e., for a multi-source M/M/1 queueing model. Moreover, we derive three approximate expressions for the average AoI for a multi-source M/G/1 queueing model having a general service time distribution. Simulation results are provided to validate the derived exact average AoI expression, to assess the tightness of the proposed approximations, and to demonstrate the AoI behavior for different system parameters.

Assessment of Spectrum Handoff Performance in Cognitive Radio Cellular Networks

Spectrum Handoff is an essential process to assure unified, effective, and robust service for cognitive users (CUs) in cognitive radio cellular networks (CRCNs). This letter models the key handoff performance measuring metrics: link maintenance probability, link failure probability, and service completion probability of a non-stationary CU and analyzes the impact of mobility parameters to investigate the characteristics and behavior of a CRCN. In addition, we present the impact of general service time distributions on service completion probability. Results show that the service time distributions have a significant impact on spectrum handoff performance metrics for cases of higher mobility.

Insensitivity of the mean field limit of loss systems under SQ(d) routeing

AbstractIn this paper, we study a large multi-server loss model under the SQ(d) routeing scheme when the service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity, giving rise to a mean field model. The fixed point of the limiting mean field equations (MFEs) was seen to be insensitive to the service time distribution in simulations, but no proof was available. While insensitivity is well known for loss systems, the models, even with state-dependent inputs, belong to the class of linear Markov models. In the context of SQ(d) routeing, the resulting model belongs to the class of nonlinear Markov processes (processes whose generator itself depends on the distribution) for which traditional arguments do not directly apply. Showing insensitivity to the general service time distributions has thus remained an open problem. Obtaining the MFEs in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFEs and then show that the MFEs have a unique fixed point that coincides with the fixed point in the exponential case, thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean field limit.

Performance analysis and stability of multiclass orbit queue with constant retrial rates and balking

In this paper, we consider a single-server retrial model with multiple classes of customers. Arrival of customers follow independent Poisson rule. A new customer, facing a busy server upon his arrival, may join the corresponding (class-dependent) orbit queue with a class-dependent probability, or leaves the system forever (balks). The orbit queues follow constant retrial rate discipline, that is, only one (oldest) orbital customer of each orbit queue makes attempts to occupy the server, in a gap of class-dependent exponential times. Within each class, service times are assumed to be independent and identically distributed (iid). We show that this setting generalizes the so-called two-way communication systems.This multiclass system with general service time distributions is analysed using regenerative approach. Necessary and sufficient stability conditions, as well as some explicit expressions for the basic steady-state probabilities, are obtained. A restricted, two-way communication model with exponential service time distributions, is analysed by matrix-analytic method. Moreover, we combine both methods to efficiently derive explicit solutions for the restricted model.An extensive simulation analysis is performed to gain deep insight into the model stability and performance. It is shown that both the simulated and exact results agree on some important measures for which analytical expressions are available, and hence establish the validity of our theoretical treatment. We numerically study the sophisticated dependence structure of the model to uncover the orbits interaction. We give further details and intuitive explanation for the system performance which complements the derived explicit expressions.

Performance of Redundancy( d ) with Identical/Independent Replicas

Queueing systems with redundancy have received considerable attention recently. The idea of redundancy is to reduce latency by replicating each incoming job a number of times and to assign these replicas to a set of randomly selected servers. As soon as one replica completes service the remaining replicas are cancelled. Most prior work on queueing systems with redundancy assumes that the job durations of the different replicas are independent and identically distributed (i.i.d.), which yields insights that can be misleading for computer system design. In this article, we develop a differential equation, using the cavity method, to assess the workload and response time distribution in a large homogeneous system with redundancy without the need to rely on this independence assumption. More specifically, we assume that the duration of each replica of a single job is identical across the servers and follows a general service time distribution. Simulation results suggest that the differential equation yields exact results as the system size tends to infinity and can be used to study the stability of the system. We also compare our system to the one with i.i.d. replicas and show the similarity in the analysis used for independent, respectively, identical replicas.