Poisson Arrival Process Research Articles

Priority Multi-Server Queueing System with Heterogeneous Customers

In this paper, we analyze a multi-server queueing system with heterogeneous customers that arrive according to a marked Markovian arrival process. Customers of two types differ in priorities and parameters of phase type distribution of their service time. The queue under consideration can be used to model the processes of information transmission in telecommunication networks in which often the flow of information is the superposition of several types of flows with correlation of inter-arrival times within each flow and cross-correlation. We define the process of information transmission as the multi-dimensional Markov chain, derive the generator of this chain and compute its stationary distribution. Expressions for computation of various performance measures of the system, including the probabilities of loss of customers of different types, are presented. Output flow from the system is characterized. The presented numerical results confirm the high importance of account of correlation in the arrival process. The values of important performance measures for the systems with the correlated arrival process are essentially different from the corresponding values for the systems with the stationary Poisson arrival process. Measurements in many real world systems show poor approximation of real flows by such an arrival process. However, this process is still popular among the telecommunication engineers due to the evident existing gap between the needs of adequately modeling the real life systems and the current state of the theory of algorithmic methods of queueing theory.

Quality control for a detected an appropriate queue from K‐independent M/M/C/N queueing models

AbstractIn this paper, we consider a system of K‐independent Markovian queues such that each one of them has a Poisson arrival process and exponential service time. We assume that every server has some characteristics such as the speed of the service performance or the service cost. To find an appropriate queue, which meets customer needs for the service performance, we present a new approach that gives a suitable decision to choose an appropriate queue from our system. This allows the customer to deal with minimum cost and faster server under steady state. We solve an interesting discrete stochastic optimization problem where the paid cost by the customer is bounded by a Gaussian distribution. Using these hypotheses, we perform a simulation study by generating the paid cost random values and choosing the minimum value between them. This minimum cost gives the highest service rate, which is used to obtain the optimum values of the system effectiveness measures.

A lattice method for jump-diffusion process applied to transmission expansion investments under demand and distributed generation uncertainties

Strategic decision making for rare events is considered to be an arduous course of actions as their impacts cannot accurately be modeled. Transmission of electrical power is a major challenge in this day and age due to the prevalent of rare events. Decision makers of generation units, distribution utilities, and transmission networks do not often cooperate, which creates severe uncertainties for all players. Growth of demand for electricity and installation or removal of distributed generation (DG), considered to be a rare event, are among uncertainties encountered by transmission network owners. Expansion decisions for transmission lines should be strategically executed because installations of DGs may create a stranded cost for transmission owners. In this study, we propose a real options framework that quantifies the values of transmission investments under demand and DG uncertainties to guide decision makers of transmission companies regarding how to adapt their expansions decisions. We model demand uncertainty as a geometric Brownian motion process and DG uncertainty as a Poisson arrival process. We devise a computationally-efficient lattice diagram to discretize both processes in a single grid, which can also be employed to model other types of rare events in various sectors. The proposed framework is demonstrated on a realistic transmission network. Numerical study shows that our proposed lattice model is computationally superior over existing diagrams. As for managerial insights, it shows that depending on the locations of DG installations, DG penetration may not reduce the value of a transmission network. If the center at which a DG is installed has a large-capacity local generator, the installation may not undervalue the transmission network.

Optimal Pricing Analysis of Computer Networks Based on a Queueing System With Retrial Mechanism

We consider a single-server retrial queue with a Poisson arrival process and exponential service times, where the server is unreliable. Assume there is no waiting space in front of the server and the customer who finds the server unavailable joins an orbit to access the server some time later. We discuss two types of customers’ retrial behavior. One is that each customer in the orbit seeks for services independently and the total retrial rate of the system depends on the number of customers in the orbit. The other type of retrial discipline is called constant retrial policy and it arises from some situations in the computer and communication network where the retrial rate may be controlled by automatic mechanisms. An announced price charged by the server is imposed on customers joining the system and the actual demands for services depend on the price via a decreasing function. We investigate the system characteristics and study how the manager, whose goal is to maximize its own profit, determines the price charging joining customers. Finally, we present an application example to illustrate the obtained results and make comparisons between the two retrial policies from the perspective of customers’ expected waiting time.

Nonstationarity and Abandonment in Markovian Queues with Application to Call Centers

Both large and small firms maintain call centers to establish contact between themselves and their customers. A call center is staffed by Customer Service Representatives (CSRs). In addition to CSRs, a call center needs computers and telecommunication equipment such as an Automatic Call Distributor (ACD). If calls arrive according to a Poisson arrival process and the service times have an exponential distribution, the call center can be modeled as an M/M/s queue where s is the number of CSRs. Typical calculations include finding the number of CSRs required and finding the number of trunks lines required. However, if call center models ignore the abandonment behavior of customers, they distort information that is relevant to management. Typically, ignoring abandonment would lead to overstaffing, as fewer CSRs are actually needed, because some of the callers abandon the system. Also, nonstationarity of arrivals is highly prevalent in call centers. Green, Kolesar and Whitt plot hourly arrival rates for a financial services call center. There is significant variation in arrivals by time of day. In this paper, we model call centers as multiserver Markovian queues with both nonstationarity and abandonment. Nonstationarity is modeled by having a Poisson arrival stream with time dependent mean, which varies according to a sinusoid. Abandonment is defined in terms of an exponential patience random variable, which is the amount of time the caller would be on hold before abandoning the call. We numerically study the performance of this nonstationary M(t)/M/s+M queue with abandonment and compare its performance measures with those of the stationary M/M/s+M queue. We find that approximating a nonstationary system by a stationary system, even at low levels of nonstationarity, can lead to significant errors. Similar results in a system without abandonment, have been obtained by Green, Kolesar and Svoronos. Additionally, we find that abandonment dampens the effect of nonstationarity. Since abandonment and nonstationarity are both present together in real call centers, a real call center with a high level of abandonment behaves closer to an ideal stationary system.

Condition-Based Maintenance for Queues With Degrading Servers

The integration of condition monitoring with queueing systems to support decision making is not well explored. This paper addresses the impact of condition monitoring of the server on the system-level performance experienced by entities in a queueing system. The system consists of a queue with a single-server subject to Markovian degradation. The model assumes a Poisson arrival process with service times and repair times according to general distributions. We develop stability conditions and perform steady-state analysis to obtain performance measures (average queue length, average degradation, and so on). We propose minimizing an objective function involving four types of costs: repair, catastrophic failure, quality, and holding. The queue performance measures derived from steady-state analysis are benchmarked and compared to those from a discrete event simulation model. After verifying the queuing model, a sensitivity analysis is performed to determine the relationships between system performance and model parameters. Results indicate that the total cost function is convex and, thus, subject to an optimal repair policy. The model is sensitive to service time, quality costs, and failure costs for late-stage policy repairs decisions and sensitive to expected repair times and repair costs for early stage policy repair decisions. Note to Practitioners —We propose condition-based maintenance for queues with degrading servers. The proposed model has been verified in simulation using degradation data of tools in a manufacturing facility. In order to implement, one must first model the underlying degradation process using the data from condition monitoring. We can get service and repair time distributions using data from the plant floor. Then, we need reasonable estimates of costs. These include costs of holding, quality, repair, and failure. They can be relative, prioritized or zero. The long-run average cost per unit time can then be evaluated. Then, the model provides the degradation state of the server at which maintenance is the most cost efficient. The degradation data can be used to continually obtain the optimal state of degradation at which the server can be taken offline with minimal costs.

Nonparametric Estimation of Service Time Characteristics in Infinite-Server Queues with Nonstationary Poisson Input

This paper provides a mathematical framework for estimation of the service time distribution and expected service time of an infinite-server queueing system with a nonhomogeneous Poisson arrival process in the case of partial information, where only the numbers of busy servers are observed over time. The problem is reduced to a statistical deconvolution problem, which is solved by using Laplace transform techniques and kernels for regularization. Upper bounds on the mean squared error of the proposed estimators are derived. Some concrete simulation experiments are performed to illustrate how the method can be applied and provide some insight in the practical performance.

Estimation of Poisson Arrival Processes Under Linear Models

In this paper, we consider the problem of estimating the parameters of a Poisson arrival process, where the intensity function is assumed to lie in the span of a known basis. Our goal is to estimate the basis expansions coefficients given a realization of this process. We establish novel guarantees concerning the accuracy achieved by the maximum likelihood estimate. Our initial result is near-optimal, with the exception of an undesirable dependence on the dynamic range of the intensity function. We then show how to remove this dependence through a process of “noise regularization,” which results in an improved bound under our analysis. We conjecture that a similar guarantee should be possible when using a more direct (deterministic) regularization scheme. We conclude with a discussion of practical applications and an empirical examination of the proposed regularization schemes.

The Mean-field Behavior of Processor Sharing Systems with General Job Lengths Under the SQ(d) Policy

In this paper, we derive the mean-field behavior of empirical distributions of large systems that consist of N (large) identical parallel processor sharing servers with Poisson arrival process having intensity Nλ and generally distributed job lengths under the randomized SQ(d) load balancing policy. Under this policy, an arrival is routed to the server with the least number of progressing jobs among d randomly chosen servers. The mean-field is then used to approximate the statistical properties of the system. In particular, we show that in the limit as N grows, individual servers are statistically independent of others (propagation of chaos) and more importantly, the equilibrium point of the mean-field is insensitive to the job length distributions. This has important engineering relevance for the robustness of such routing policies that are often used in web server farms. We use a measure-valued process approach and martingale techniques to obtain our results. We also provide numerical results to support our analysis.

Asymptotic correlation structure of discounted Incurred But Not Reported claims under fractional Poisson arrival process

This paper studies the joint moments of a compound discounted renewal process observed at different times with each arrival removed from the system after a random delay. This process can be used to describe the aggregate (discounted) Incurred But Not Reported claims in insurance and also the total number of customers in an infinite server queue. It is shown that the joint moments can be obtained recursively in terms of the renewal density, from which the covariance and correlation structures are derived. In particular, the fractional Poisson process defined via the renewal approach is also considered. Furthermore, the asymptotic behaviour of covariance and correlation coefficient of the aforementioned quantities is analyzed as the time horizon goes to infinity. Special attention is paid to the cases of exponential and Pareto delays. Some numerical examples in relation to our theoretical results are also presented.

On the Inter-Departure Times in <inline-formula> <tex-math notation="LaTeX">$M/\widetilde{D}/1/{B}_{on}$ </tex-math> </inline-formula> Queue With Queue-Length Dependent Service and Deterministic/Exponential Vacations

We derive the distribution of inter-departure times of a finite-buffer single-server queue with Poisson arrival process and queue-length-dependent service times, where the server goes to vacation if either the queue is emptied or a limited number $(R_{1})$ of packets are served, whichever occurs first, in the current busy period. We consider two types of vacation distributions: 1) deterministic and 2) exponential. Queue-length distribution at embedded points is derived first, then, the distribution and variance of inter-departure times are derived, for both types of vacations. The simulation results are in good agreement with the derived analytical results. The above framework would be useful at the receiver in modeling and analyzing the jitter and the waiting time of time-division multiplexing (TDM) emulated packets in TDM over packet-switched network (TDM over PSN) technology as a function of a buffer size.

Performance analysis of an unreliable M/G/1 retrial queue with two-way communication

Efficient use of call center operators through technological innovations more often come at the expense of added operation management issues. In this paper, the stationary characteristics of an $M/G/1$ retrial queue is investigated where the single server, subject to active failures, primarily attends incoming calls and directs outgoing calls only when idle. The incoming calls arriving at the server follow a Poisson arrival process, while outgoing calls are made in an exponentially distributed time. On finding the server unavailable (either busy or temporarily broken down), incoming calls intrinsically join the virtual orbit from which they re-attempt for service at exponentially distributed time intervals. The system stability condition along with probability generating functions for the joint queue length distribution of the number of calls in the orbit and the state of the server are derived and evaluated numerically in the context of mean system size, server availability, failure frequency and orbit waiting time.

Optimal Sleeping Mechanism for Multiple Servers With MMPP-Based Bursty Traffic Arrival

A fundamental problem in green communications and networking is the operation of servers (routers or base stations) with sleeping mechanism to optimize energy-delay tradeoffs. This problem is very challenging when considering realistic bursty (non-Poisson) traffic. We prove for the first time that the optimal structure of such a sleeping mechanism for multiple servers when the arrival of jobs is modeled by a bursty Markov-modulated Poisson process (MMPP). It is shown that the optimal operation, which determines the number of active (or sleeping) servers dynamically, is hysteretic and monotone, and hence, it is a queue-threshold-based policy. This letter settles a conjecture in the literature that the optimal sleeping mechanism for a single server with interrupted Poisson arrival process, which can be treated as a special case of MMPP, is queue-threshold-based. The exact thresholds are given by numerically solving the Markov decision process.

Cooperation and sharing costs in a tandem queueing network

We consider a tandem network of queues with a Poisson arrival process to the first queue. Service times are assumed to be exponential. In cases where they are not, we additionally assume a processor sharing service discipline in all servers. Consecutive servers may cooperate by pooling resources which leads to the formation of a single combined server that satisfies the aggregated service demands with a greater service rate. On this basis we define a cooperative game with transferable utility, where the cost of a coalition is the steady-state mean total number of customers in the system formed by its members. We show that the game is subadditive, leading to full cooperation being socially optimal. We then show the non-emptiness of the core, despite the characteristic function being neither monotone, nor concave. Finally, we derive several well-known solution concepts, including the Shapley value, the Banzhaf, value and the nucleolus, for the case where servers have equal mean service demands. In particular, we show that all three values coincide in this case.

Statistical Modeling of Single-Photon Avalanche Diode Receivers for Optical Wireless Communications

In this paper, a comprehensive analytical approach is presented for modeling the counting statistics of active quenching and passive quenching single-photon avalanche diode (SPAD) detectors. It is shown that, unlike ideal photon counting receiver for which the detection process is described by a Poisson arrival process, photon counts in practical SPAD receivers do not follow a Poisson distribution and are highly affected by the dead time caused by the quenching circuit. Using the concepts of renewal theory, the exact expressions for the probability distribution and moments (mean and variance) of photocounts in the presence of dead time are derived for both active quenching and passive quenching SPADs. The derived probability distributions are validated through Monte Carlo simulations and it is demonstrated that the moments match with the existing empirical models for the moments of SPAD photocounts. Furthermore, an optical communication system with on–off keying and binary pulse position modulation is considered and the bit error performance of the system for different dead time values and background count levels is evaluated.

Aggregating courier deliveries

AbstractWe consider the problem of efficiently scheduling deliveries by an uncapacitated courier from a central location under online arrivals. We consider both adversary‐controlled and Poisson arrival processes. In the adversarial setting we provide a randomized (3βΔ/2δ − 1)‐competitive algorithm, where β is the approximation ratio of the traveling salesman problem, δ is the minimum distance between the central location and any customer, and Δ is the length of the optimal traveling salesman tour overall customer locations and the central location. We provide instances showing that this analysis is tight. We also prove a 1 + 0.271Δ/δ lower‐bound on the competitive ratio of any algorithm in this setting. In the Poisson setting, we relax our assumption of deterministic travel times by assuming that travel times are distributed with a mean equal to the excursion length. We prove that optimal policies in this setting follow a threshold structure and describe this structure. For the half‐line metric space we bound the performance of the randomized algorithm in the Poisson setting, and show through numerical experiments that the performance of the algorithm is often much better than this bound.

CarrierSeq: a sequence analysis workflow for low-input nanopore sequencing

BackgroundLong-read nanopore sequencing technology is of particular significance for taxonomic identification at or below the species level. For many environmental samples, the total extractable DNA is far below the current input requirements of nanopore sequencing, preventing “sample to sequence” metagenomics from low-biomass or recalcitrant samples.ResultsHere we address this problem by employing carrier sequencing, a method to sequence low-input DNA by preparing the target DNA with a genomic carrier to achieve ideal library preparation and sequencing stoichiometry without amplification. We then use CarrierSeq, a sequence analysis workflow to identify the low-input target reads from the genomic carrier. We tested CarrierSeq experimentally by sequencing from a combination of 0.2 ng Bacillus subtilis ATCC 6633 DNA in a background of 1000 ng Enterobacteria phage λ DNA. After filtering of carrier, low quality, and low complexity reads, we detected target reads (B. subtilis), contamination reads, and “high quality noise reads” (HQNRs) not mapping to the carrier, target or known lab contaminants. These reads appear to be artifacts of the nanopore sequencing process as they are associated with specific channels (pores).ConclusionBy treating sequencing as a Poisson arrival process, we implement a statistical test to reject data from channels dominated by HQNRs while retaining low-input target reads.

AN ANALYSIS OF QOS IN SDN-BASED NETWORK BY QUEUING MODEL

The analytical approach for analysis the quality of service characteristics of Software-Defined Networking is proposed in the paper. Software-Defined Networking is perspective concept that aimed to improving the performance and efficiency of computing networks. Current complexity of networks based this concept requires define the adequate methods for the calculation work characteristics. The model characterized by a Poisson arrival process with parameter λ and service time characterized by hyper-exponential distribution with r phases proposed to use for analysis QoS. Performance analysis using the proposed queuing model indicates that the model is accurate. Such QoS characteristics as maximum flow intensity that can be serving on switch and on controller, average packet sojourn time on single serve elements, delay and CP utilization of serve elements are calculated.

Optimal Periodic Dividend Strategies for Spectrally Positive Lévy Risk Processes With Fixed Transaction Costs

We consider the general class of spectrally positive Levy risk processes, which are appropriate for businesses with continuous expenses and lump sum gains whose timing and sizes are stochastic. Motivated by the fact that dividends are paid periodically in real life, we study periodic dividend strategies whereby dividend decisions are made according to a Poisson arrival process. In this paper, we investigate the impact of fixed transaction costs on the optimal periodic dividend strategy, and show that a periodic (b u , b l ) strategy is optimal. Such a strategy leads to lump sum dividends that bring the surplus back to b l as long as it is no less than b u at a dividend decision time. The expected present value of dividends (net of transaction costs) is provided explicitly with the help of scale functions. Results are illustrated.

Rare-event analysis of mixed Poisson random variables, and applications in staffing

A common assumption when modeling queueing systems is that arrivals behave like a Poisson process with constant parameter. In practice, however, call arrivals are often observed to be significantly overdispersed. This motivates that in this paper we consider a mixed Poisson arrival process with arrival rates that are resampled every N−α time units, where α>0 and N a scaling parameter.In the first part of the paper we analyze the asymptotic tail distribution of this doubly stochastic arrival process. That is, for large N and i.i.d. arrival rates X1,…,XN, we focus on the evaluation of the probability that the scaled number of arrivals exceeds Na, PN(a)≔PPoisNX¯Nα⩾Na,withX¯N≔1N∑i=1NXi.The logarithmic asymptotics of PN(a) are easily obtained from previous results; we find constants rP and γ such that N−γlogPN(a)→−rP as N→∞. Relying on elementary techniques, we then derive the exact asymptotics of PN(a): For α<13 and α>3 we identify (in closed-form) a function P˜N(a) such that PN(a)∕P˜N(a) tends to 1 as N→∞. For α∈[13,12) and α∈[2,3) we find a partial solution in terms of an asymptotic lower bound. For the special case that the Xis are gamma distributed, we establish the exact asymptotics across all α>0. In addition, we set up an asymptotically efficient importance sampling procedure that produces reliable estimates at low computational cost.The second part of the paper considers an infinite-server queue assumed to be fed by such a mixed Poisson arrival process. Applying a scaling similar to the one in the definition of PN(a), we focus on the asymptotics of the probability that the number of clients in the system exceeds Na. The resulting approximations can be useful in the context of staffing. Our numerical experiments show that, astoundingly, the required staffing level can actually decrease when service times are more variable.