Service Time Distribution Research Articles

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.

Divide and Conquer: A Hygienic, Efficient, and Reliable Assembly Line for Housekeeping

Problem definition: This work focuses on the hotel housekeeping process. In a field study, a possible channel of disease transmission between consecutive guests in hotel rooms is revealed. In order to prevent the transmission, an innovative assembly-line housekeeping method is developed. Academic/practical relevance: The transmission of infectious diseases during hotel stays (e.g., by touching unclean towels or bed linens) has been reported globally. Under the current COVID-19 pandemic, having contact with saliva or mucus left by an infected person could cause infection. The standard housekeeping process used by the majority of hotels leaves a channel for new towels and bed linens in refreshed rooms to be contaminated by bacteria or viruses from used towels and bed linens. Eliminating the contamination channel and preventing disease transmission are crucial for protecting the health and safety of hotel guests, especially under a disease outbreak such as the current COVID-19 pandemic. Methodology: The research was conducted during a field study at a hotel. To design the assembly-line process, the service time distribution of each housekeeping operational step is characterized using data collected from the practice at hundreds of hotel rooms. An optimization model is proposed to optimize the operation. Through a pilot test, the performance of the assembly-line and the traditional housekeeping methods is compared. Results: The pilot test results show that the assembly-line housekeeping method has the potential to improve not only hygienic standards but also, labor efficiency and service quality (error rate). Managerial implications: The outbreak of the COVID-19 pandemic draws tremendous public attention on disease transmission and public hygiene. The principle of the assembly-line method (i.e., eliminating contamination channels through teamwork operational design) can be applied to not only hotel housekeeping practices but also, many other service settings. It leads to hygienic, efficient, and reliable operations, at no additional cost.

Sensitivity of mean-field fluctuations in Erlang loss models with randomized routing

AbstractIn this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; an incoming job is routed to a server if it has the lowest occupancy amongst d (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of d) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$ , $\sigma\in\mathbb{R}_+$ and $\beta\in\mathbb{R}$ , we establish functional central limit theorems for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein–Uhlenbeck process whose mean and variance depend on the mean field of the considered model. Using this, we obtain approximations to the blocking probabilities for large N, where we can precisely estimate the accuracy of first-order approximations.

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.

Learning Deep Generative Models for Queuing Systems

Modern society is heavily dependent on large scale client-server systems with applications ranging from Internet and Communication Services to sophisticated logistics and deployment of goods. To maintain and improve such a system, a careful study of client and server dynamics is needed – e.g. response/service times, aver-age number of clients at given times, etc. To this end, one traditionally relies, within the queuing theory formalism,on parametric analysis and explicit distribution forms.However, parametric forms limit the model’s expressiveness and could struggle on extensively large datasets. We propose a novel data-driven approach towards queuing systems: the Deep Generative Service Times. Our methodology delivers a flexible and scalable model for service and response times. We leverage the representation capabilities of Recurrent Marked Point Processes for the temporal dynamics of clients, as well as Wasserstein Generative Adversarial Network techniques, to learn deep generative models which are able to represent complex conditional service time distributions. We provide extensive experimental analysis on both empirical and synthetic datasets, showing the effectiveness of the proposed models

A MAP/PH(1), PH(2)/2 Production Inventory System with Multiple Servers and Varying Service Rates

The paper comprises an (s,S) production inventory facility in which multiple vacations and varying service rates are considered for the multiple servers. The arrival of customers constitutes a Markovian Arrival Process (MAP) and service completion times follow the phase type distributions with representations ( α, S ) and ( β, T ) respectively. Manufacturing needs to begin at the moment when the stock position falls to s. Service is offered at a lower rate if the stock level ranges from 0 to s and the service time distribution has the representations (α, η 1 S) and (β, η 2 T ), 0 < η 1, η 2 < 1 respectively. If the stock level reaches the maximum S, production is ceased to operate. 1-limited service policy, Bernoulli service schedule, and Exhaustive service disciplines are considered for the servers. A suitable cost function is outlined as per performance assessment. The impact of negative correlated inter arrival times on cost function is illustrated. Also, a relative study of expected cost function on different service modes is presented.

Assigning Priorities (or Not) in Service Systems with Nonlinear Waiting Costs

For a queueing system with multiple customer types differing in service-time distributions and waiting costs, it is well known that the cµ-rule is optimal if costs for waiting are incurred linearly with time. In this paper, we seek to identify policies that minimize the long-run average cost under nonlinear waiting cost functions within the set of fixed priority policies that only use the type identities of customers independently of the system state. For a single-server queueing system with Poisson arrivals and two or more customer types, we first show that some form of the cµ-rule holds with the caveat that the indices are complex, depending on the arrival rate, higher moments of service time, and proportions of customer types. Under quadratic cost functions, we provide a set of conditions that determine whether to give priority to one type over the other or not to give priority but serve them according to first-come, first-served (FCFS). These conditions lead to useful insights into when strict (and fixed) priority policies should be preferred over FCFS and when they should be avoided. For example, we find that, when traffic is heavy, service times are highly variable, and the customer types are not heterogenous, so then prioritizing one type over the other (especially a proportionally dominant type) would be worse than not assigning any priority. By means of a numerical study, we generate further insights into more specific conditions under which fixed priority policies can be considered as an alternative to FCFS. This paper was accepted by Baris Ata, stochastic models and simulation.

M/PH/1 queueing model with re-servicing

In this paper, we analyze an M/PH/1 queuing system with optional re-service. In this queueing system the interarrival time distribution is exponential with mean service time and re-service time distribution follows continuous time phase-type distribution and each customer will be re-serviced with probability ‘p’. We have obtained the expressions of various system performance measures and maximum likelihood method using E-M algorithm and Bayesian procedure based on MCMC method were proposed to estimate system parameters. Numerical illustration of various special cases of the queueing system was carried out using simulated data sets.

Batch sojourn time in the processor sharing queue with geometric batch size

In this paper, we analyze the sojourn time of an entire batch in a processor sharing queue, where geometrically distributed batches arrive according to a Poisson process and individual jobs require exponentially distributed service times. By conditioning on the number of jobs in the queue and the number of jobs in a tagged batch, we establish recurrence relations for conditional sojourn times, which subsequently allow us to derive a partial differential equation for an associated bivariate generating function. This equation involves an unknown generating function, whose series expansion can be computed by solving an infinite lower triangular linear system. Once this unknown function is determined, we determine the Laplace transform and the mean value of the sojourn time of a batch in the system.

Synergy via Redundancy: Adaptive Replication Strategies and Fundamental Limits

The maximum possible throughput (or the rate of job completion) of a multi-server system is typically the sum of the service rates of individual servers. Recent work shows that launching multiple replicas of a job and canceling them as soon as one copy finishes can boost the throughput, especially when the service time distribution has high variability. This means that redundancy can, in fact, create synergy among servers such that their overall throughput is greater than the sum of individual servers. This work seeks to find the fundamental limit of the throughput boost achieved by job replication and the optimal replication policy to achieve it. While most previous works consider upfront replication policies, we expand the set of possible policies to delayed launch of replicas. The search for the optimal adaptive replication policy can be formulated as a Markov Decision Process, using which we propose two myopic replication policies, MaxRate and AdaRep, to adaptively replicate jobs. In order to quantify the optimality gap of these and other policies, we derive upper bounds on the service capacity, which provide fundamental limits on the throughput of queueing systems with redundancy.

On the joint distribution of an infinite-buffer discrete-time batch-size-dependent service queue with single and multiple vacations

ABSTRACT Due to the widespread applicability of discrete-time queues in wireless networks or telecommunication systems, this paper analyzes an infinite-buffer batch-service queue with single and multiple vacations where customers/messages arrive according to the Bernoulli process and service time varies with the batch-size. The foremost focal point of this analysis is to get the complete joint distribution of queue length and server content at the service completion epoch, for which first the bivariate probability generating function has been derived. We also acquire the joint distribution at an arbitrary slot. We also provide several marginal distributions and performance measures for the utilization of the vendor. During the transmission of data through a particular communication channel, high transmission error may take place due to several factors. For this reason, one may skip the transmission through that particular channel. However the discrete phase-type distribution plays a noteworthy role to control this transmission error which eventually motivates us to include a numerical example where service time distribution follows discrete phase-type distribution. A comparison between batch-size-dependent and -independent services has been drawn through the graphical representation of some performance measures and total system cost.

MATHEMATICAL DELAY MODEL BASED ON SYSTEMS WITH HYPERERLANGIAN AND ERLANGIAN DISTRIBUTIONS

Context. Studies of G/G/1 systems in queuing theory are relevant because such systems are of interest for analyzing the delay of data transmission systems. At the same time, it is impossible to obtain solutions for the delay in the final form in the general case for arbitrary laws of distribution of the input flow and service time. Therefore, it is important to study such systems for particular cases of input distributions. We consider the problem of deriving a solution for the average queue delay in a closed form for two systems with ordinary and shifted hypererlangian and erlangian input distributions.&#x0D; Objective. Obtaining a solution for the main characteristic of the system – the average delay of requests in the queue for two queuing systems of the G/G/1 type with ordinary and with shifted hypererlangian and erlangian input distributions.&#x0D; Method. To solve this problem, we used the classical method of spectral decomposition of the solution of the Lindley integral equation. This method allows to obtaining a solution for the average delay for systems under consideration in a closed form. The method of spectral decomposition of the solution of the Lindley integral equation plays an important role in the theory of systems G/G/1. For the practical application of the results obtained, the well-known method of moments of probability theory is used.&#x0D; Results. For the first time, spectral expansions of the solution of the integral Lindley equation for two systems are obtained, with the help of which calculation formulas for the average delay in a queue in a closed form are derived. Thus, mathematical models of queuing delay for these systems have been built.&#x0D; Conclusions. These formulas expand and supplement the known queuing theory formulas for the average delay G/G/1 systems with arbitrary laws distributions of input flow and service time. This approach allows us to calculate the average delay for these systems in mathematical packages for a wide range of traffic parameters. In addition to the average delay, such an approach makes it possible to determine also moments of higher orders of waiting time. Given the fact that the packet delay variation (jitter) in telecommunications is defined as the spread of the delay from its average value, the jitter can be determined through the variance of the delay.

Efficient Replication for Fast and Predictable Performance in Distributed Computing

Master-worker distributed computing systems use task replication to mitigate the effect of slow workers on job compute time. The master node groups tasks into batches and assigns each batch to one or more workers. We first assume that the batches do not overlap. Using majorization theory, we show that a balanced replication of batches minimizes the average job compute time for a general class of service time distributions. We then show that the balanced assignment of non-overlapping batches achieves a lower average job compute time than the overlapping schemes proposed in the literature. Next, we derive the optimum redundancy level as a function of the task service time distribution. We show that the redundancy level that minimizes the average job compute time may not coincide with the redundancy level that maximizes job compute time predictability. Therefore, there is a trade-off in optimizing the two metrics. By running experiments on Google cluster traces, we observe that redundancy can reduce the job compute time by one order of magnitude. The optimum level of redundancy depends on the distribution of task service time.

Appointment scheduling for multi-stage sequential service systems with limited distributional information

We study a multi-stage appointment scheduling problem with limited distributional information of stochastic service times, and formulate the distributionally robust (DR) model. In our studied problem, only the means and supports of service times are known to the planner, and the planner has to determine a job allowance for each appointment in the first stage. The objective is to minimize the maximum total expected weighted costs of customers’ waiting times and service providers’ idle times and overtimes over multiple stages under the worst-case distribution of service times. For the DR multi-stage appointment scheduling model, we first establish linear relationships among performance indicators (i.e., waiting times, idle times and overtimes), and then develop a cutting-plane approach to solve it. To overcome the difficulty of implementing the cutting-plane approach, we reformulate the inner maximization problem in the DR model as a mixed-integer linear program. Finally, computational experiments are conducted to evaluate the computational and simulational performance of the DR model, investigate the structure of the optimal schedule and examine the efficiency of some potential sequencing heuristics.

Minimum Achievable Peak Age of Information Under Service Preemptions and Request Delay

There is a growing interest in analysing freshness of data in networked systems. Age of Information (AoI) has emerged as a relevant metric to quantify this freshness at a receiver, and minimizing this metric for different system models has received significant research attention. However, a fundamental question remains: what is the minimum achievable AoI in any single-server-single-source queuing system for a given service-time distribution? We address this question for the average peak AoI (PAoI) statistic by considering generate-at-will source model, service preemptions, and request delays. Our main result is on the characterization of the minimum achievable average PAoI, and we show that it is achieved by a fixed-threshold policy among the set of all causal policies. We use the characterization to provide necessary and sufficient condition for preemptions to be beneficial for a given service-time distribution. Our numerical results, obtained using well-known distributions, demonstrate that the heavier the tail of a distribution the higher the performance gains of using preemptions.

Steady‐state analysis of load balancing with Coxian‐2 distributed service times

AbstractThis paper studies load balancing for many‐server (N servers) systems. Each server has a buffer of size b − 1, and can have at most one job in service and b − 1 jobs in the buffer. The service time of a job follows the Coxian‐2 distribution. We focus on steady‐state performance of load balancing policies in the heavy traffic regime such that the normalized load of system is λ = 1 − N−α for 0 &lt; α &lt; 0.5. We identify a set of policies that achieve asymptotic zero waiting. The set of policies include several classical policies such as join‐the‐shortest‐queue (JSQ), join‐the‐idle‐queue (JIQ), idle‐one‐first (I1F) and power‐of‐d‐choices (Po d) with d = O(Nα log N). The proof of the main result is based on Stein's method and state space collapse. A key technical contribution of this paper is the iterative state space collapse approach that leads to a simple generator approximation when applying Stein's method.

Analysis of a k-Stage Bulk Service Queuing System with Accessible Batches for Service

In most of the service systems considered so far in queuing theory, no fresh customer is admitted to a batch undergoing service when the number in the batch is less than a threshold. However, a few researchers considered the case of customers accessing ongoing service batch, irrespective of how long service was provided to that batch. A queuing system with a different kind of accessibility that relates to a real situation is studied in the paper. Consider a single server queuing system in which the service process comprises of k stages. Customers can enter the system for service from a node at the beginning of any of these stages (provided the pre-determined maximum service batch size is not reached) but cannot leave the system after completion of service in any of the intermediate stages. The customer arrivals to the first node occur according to a Markovian Arrival Process (MAP). An infinite waiting room is provided at this node. At all other nodes, with finite waiting rooms (waiting capacity cj,2≤j≤k), customer arrivals occur according to distinct Poisson processes with rates λj,2≤j≤k. The service is provided according to a general bulk service rule, i.e., the service process is initiated only if at least a customers are present in the queue at node 1 and the maximum service batch size is b. Customers can join for service from any of the subsequent nodes, provided the number undergoing service is less than b. The service time distribution in each phase is exponential with service rate μjm, which depends on the service stage j,1≤j≤k, and the size of the batch m,a≤m≤b. The behavior of the system in steady-state is analyzed and some important system characteristics are derived. A numerical example is presented to illustrate the applicability of the results obtained.

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.

Zero Queueing for Multi-Server Jobs

Cloud computing today is dominated by multi-server jobs. These are jobs that request multiple servers simultaneously and hold onto all of these servers for the duration of the job. Multi-server jobs add a lot of complexity to the traditional one-server-per-job model: an arrival might not "fit'' into the available servers and might have to queue, blocking later arrivals and leaving servers idle. From a queueing perspective, almost nothing is understood about multi-server job queueing systems; even understanding the exact stability region is a very hard problem. In this paper, we investigate a multi-server job queueing model under scaling regimes where the number of servers in the system grows. Specifically, we consider a system with multiple classes of jobs, where jobs from different classes can request different numbers of servers and have different service time distributions, and jobs are served in first-come-first-served order. The multi-server job model opens up new scaling regimes where both the number of servers that a job needs and the system load scale with the total number of servers. Within these scaling regimes, we derive the first results on stability, queueing probability, and the transient analysis of the number of jobs in the system for each class. In particular we derive sufficient conditions for zero queueing. Our analysis introduces a novel way of extracting information from the Lyapunov drift, which can be applicable to a broader scope of problems in queueing systems.

A Service System with Packing Constraints: Greedy Randomized Algorithm Achieving Sublinear in Scale Optimality Gap

A service system with multiple types of arriving customers is considered. There is an infinite number of homogeneous servers. Multiple customers can be placed for simultaneous service into one server, subject to general packing constraints. The service times of different customers are independent even if they are served simultaneously by the same server; the service time distribution depends on the customer type. Each new arriving customer is placed for service immediately into either an occupied server, that is, one already serving other customers, as long as packing constraints are not violated or into an empty server. After service completion, each customer leaves its server and the system. The basic objective is to minimize the number of occupied servers in steady state. We study a greedy random (GRAND) placement (packing) algorithm, introduced in our previous work. This is a simple online algorithm that places each arriving customer uniformly at random into either one of the already occupied servers that can still fit the customer or one of the so-called zero servers, which are empty servers designated to be available to new arrivals. In our previous work, a version of the algorithm, labeled GRAND(aZ), is considered, in which the number of zero servers is aZ with Z being the current total number of customers in the system and positive a being an algorithm parameter. GRAND(aZ) is shown in our previous work to be asymptotically optimal in the following sense: (a) the steady-state optimality gap grows linearly in the system scale r (the mean total number of customers in service), that is, as c(a)r for some positive c(a), and (b) c(a) vanishes as a goes to zero. In this paper, we consider the GRAND(Z p ) algorithm, in which the number of zero servers is Z p , where p &lt; 1 is a fixed parameter, sufficiently close to 1. We prove the asymptotic optimality of GRAND(Z p ) in the sense that the steady-state optimality gap is sublinear in the system scale r. This is a stronger form of asymptotic optimality than that of GRAND(aZ).