Exponential Service Times Research Articles

Age-Upon-Decisions Minimizing Scheduling in Internet of Things: To Be Random or To Be Deterministic?

In this article, we consider an Internet of Things (IoT) system in which a sensor delivers updates to a monitor with exponential service time and first-come-first-served (FCFS) discipline. We investigate the freshness of the received updates and propose a new metric termed as age upon decisions (AuD), which is defined as the time elapsed from the generation of each update to the epoch it is used to make decisions (e.g., estimations, inferences, and controls). Within this framework, we aim at improving the freshness of updates at decision epochs by scheduling the update arrival process and the decision-making process. The theoretical results show that: 1) when the decisions are made according to a Poisson process, the average AuD is independent of decision rate and will be minimized if the arrival process is periodic (i.e., deterministic); 2) when both the decision process and the arrival process are periodic, the average AuD is larger, but decreases with decision rate to, the average AuD of the corresponding system with the Poisson decisions (i.e., random); and 3) when both the decision process and the arrival process are periodic, the average AuD can be further decreased by optimally controlling the offset between the two processes. For practical IoT systems, therefore, it is suggested to employ periodic arrival processes and random decision processes. Nevertheless, making the periodical updates and decisions with properly controlled offset is also a promising solution, if the timing information of the two processes can be accessed by the monitor.

Queueing analysis for operations modeling in port logistics

Purpose The use of queueing network models was stimulated by the appearance (1975) of the exact product form solution of a class of open, closed and mixed queueing networks obeying the local balance principle and solved, a few years later, by the popular mean value analysis algorithm (1980). Since then, research efforts have been produced to approximate solutions for non-exponential services and non-pure random mechanisms in customer processing and routing. The purpose of this paper is to examine the suitability of modeling choices and solution approaches consolidated in other domains with respect to two key logistic processes in container terminals. Design/methodology/approach In particular, the analytical solution of queueing networks is assessed for the vessel arrival-departure process and the container internal transfer process with respect to a real terminal of pure transshipment. Findings Numerical experiments show the extent to which a decomposition-based approximation, under fixed or state-dependent arrival rates, may be suitable for the approximate analysis of the queueing network models. Research limitations/implications The limitation of adopting exponential service time distributions and Poisson flows is highlighted. Practical implications Comparisons with a simulation-based solution deliver numerical evidence on the companion use of simulation in the daily practice of managing operations in a finite-time horizon under complex policies. Originality/value Discussion of some open modeling issues and encouraging results provide some guidelines on future research efforts and/or suitable adaption to container terminal logistics of the large body of techniques and algorithms available nowadays for supporting long-run decisions.

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.

Control charts for traffic intensity monitoring of Markovian multiserver queues

AbstractA number of recent research studies have applied queueing theory as an approximate modeling tool to mathematically describe industrial systems, which include manufacturing, distribution, and service, for instance. Among the main observable characteristics in queues, the number of users in the system can be controlled to keep waiting times as minimal as possible. The design of efficient control charts is an attempt to monitor and control such systems. Control charts are proposed to monitor infinite queues with Markovian arrivals, exponential service times, and s identical parallel servers. The proposed charts monitor traffic intensities, which are the ratio between the arrival rate and the service rate, estimated through the number of users in the queueing system at random epochs. The effectiveness and efficiency of the proposed approaches in terms of the average run lengths are established by a comprehensive set of Monte Carlo simulations.

Unified killing mechanism in a single server queue with renewal input

Queueing systems experienced in real-life situations are very often influenced by negative arrivals which are independent of service process and cause the elimination of jobs from the system. Such a scenario occurs in computer network and telecommunication systems where an attack by a malicious virus results in the removal of some or all data files from the system. Along this direction many authors have proposed various killing processes in the past. This paper unifies different killing mechanisms into the classical single server queue having infinite capacity, where arrival occurs as renewal process with exponential service time distribution. The system is assumed to be affected by negative customers as well as disasters. The model is investigated in steady-state in a very simple and elegant way by means of supplementary variable and difference equation technique. The distribution of system-content for the positive customers is derived in an explicit form at pre-arrival and random epochs. The influence of different parameters on the system performance are also examined.

Performance evaluation of flow lines with non-identical and unreliable parallel machines and finite buffers

This paper examines serial production lines with unreliable non-identical parallel machines at each workstation and intermediate buffers with finite capacities. All machines are assumed to have exponential service times, times to failure and repair times. An efficient decomposition technique is introduced for the performance evaluation of such lines. Rather than replacing each parallel-machine workstation with an equivalent single-server workstation, the main contribution of this paper is the presentation of a direct approach to derive and apply decomposition equations directly for every parallel machine at each workstation. Experimental results indicate that such a method can provide a computationally efficient algorithm to analyse large serial unreliable multi-server production lines with a good accuracy compared against simulation and other available methods.

재시도 준개방형 대기행렬네트워크 분석

A semi-open queueing network is considered. The network of an arbitrary topology consists of a finite number of nodes operation of which is modeled by single-server queueing systems with a buffer and an exponential service time distribution. Customers arrive at the network according to a Markovian arrival process. The total number of customers, which can be processed simultaneously, is restricted by a fixed in advance threshold. If the number of customers in the network at a new customer arrival moment is less than the threshold, this customer is admitted for service. Otherwise, the customer temporarily leaves the system (it is said that the customer joins the orbit) and retries to get access to the network after the random time intervals duration of which has an exponential distribution. The capacity of the orbit is infinite. After the admission to the network, the customer starts sequential service in a random number of nodes. The choice of the first and the subsequent nodes for service is made stochastically according to a fixed stochastic vector and a transition probability matrix. The behavior of the network is described by the continuous-time Markov chain that belongs to the class of asymptotically quasi-Toeplitz Markov chains. This allows to derive a transparent condition for the ergodicity of the chain and compute its steady-state distribution. Expressions for computation of various performance measures of the network are derived. Numerical results are presented. The model can be applied, e.g., for analysis and optimization of the operation of logistics and manufacturing systems, and the wireless telecommunication networks.

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.

Stochastic Monotonicity of Markovian Multiclass Queueing Networks

Multiclass queueing networks (McQNs) extend the classical concept of the Jackson network by allowing jobs of different classes to visit the same server. Although such a generalization seems rather natural, from a structural perspective, there is a significant gap between the two concepts. Nice analytical features of Jackson networks, such as stability conditions, product–form equilibrium distributions, and stochastic monotonicity, do not immediately carry over to the multiclass framework. The aim of this paper is to shed some light on this structural gap, focusing on monotonicity properties. To this end, we introduce and study a class of Markov processes, which we call Q-processes, modeling the time evolution of the network configuration of any open, work-conservative McQN having exponential service times and Poisson input. We define a new monotonicity notion tailored for this class of processes. Our main result is that we show monotonicity for a large class of McQN models, covering virtually all instances of practical interest. This leads to interesting properties that are commonly encountered for “traditional” queueing processes, such as (i) monotonicity with respect to external arrival rates and (ii) star-convexity of the stability region (with respect to the external arrival rates); such properties are well known for Jackson networks but had not been established at this level of generality. This research was partly motivated by the recent development of a simulation-based method that allows one to numerically determine the stability region of a McQN parameterized in terms of the arrival-rates vector.

ANALISIS MODEL ANTRIAN MULTIPLE CHANNEL MULTIPLE PHASE SERVICE DALAM PROSES PEMBUATAN KARTU MAHASISWA (KTM) PADA BANK BNI UNTAD

The queue is a common situation that is common in our daily lives as consumers waited in front of the booth to get a turn to the service or service facilities. One example is the manufacture queue Student Identity Card (KTM) Tadulako in Bank BNI UNTAD. The purposes of this research is to analyze the characteristics of the making of queue Student Identity Card (KTM) Tadulako in Bank BNI of UNTAD. By calculating the probability were no queues (𝑃0 ),the average customer in the queue (𝐿𝑎),the average customer in the system (𝐿𝑠 ),the average waiting time for customers in the queue (Wa ),the average time waiting for customers in the system (W𝑠 ). The results showed that the model type of queue that is used in the manufacture of Student Card (KTM) Tadulako in Bank BNI UNTAD is a model of multiple channels-multiple phase (M / M / S) with a Poisson arrival pattern distribution and exponential distribution service time. The analysis shows the performance characteristics of a queuing system manufacture KTM BANK BNI of UNTAD can be said to have been effective, because of the steady state in each stage. On Wednesday, the value of (𝑃0 ) at CS is 0,145 and the teller is 0.8879,for the value of (𝐿𝑎) in CS is 1.8821 and the teller customer is a customer 0.0004 tothe value (𝐿𝑠 ) 3.4948is the customer, to the value of (Wa ) on CS is 33.5373 minutes and the teller was 0.0075 minutes, to the value of (W𝑠 ) is 74.1952 minutes. On Thursday, the value of (𝑃0 ) on CS was 0.121 and the teller is 0.8837,for the value of (𝐿𝑎) in CS is 2.4998 and the teller customer is a customer 0.0005 tothe value (𝐿𝑠 ) 4.1918is the customer, to the value of (Wa ) on CS is 42.4061 minutes and the teller was 0.0081 minutes, to the value of (W𝑠 ) is 71.1808 minutes.Keywords : Exponential Distribution, Poisson Distribution, Queue Multi Phase

On the power series approximations of a structured batch arrival two-class retrial system with weighted fair orbit queues

We provide power series approximations for a structured batch arrival single server retrial system with probabilistic priority and two infinite capacity weighted fair orbit queues, under a queue-based constant retrial policy. We consider both exponential and arbitrary distributed service times. In both cases we obtain power series expansions of the generating functions of the stationary joint orbit queue-length distributions, and provide a recursive approach to calculate their coefficients. Numerical results show the accuracy of our approach, and useful insights in the system performance.

Optimal Dynamic Allocation of Collaborative Servers in Two Station Tandem Systems

We consider two-stage tandem queueing systems with one dedicated server in each station and a flexible server that can serve both stations. We assume exponential service times, linear holding costs accrued by jobs present in the system, and a collaborative work discipline. We seek optimal server allocation strategies for systems without external arrivals (clearing systems). When the combined rate of collaborating servers is less than the sum of their individual rates (partial collaboration), we identify conditions under which the optimal server allocation strategy is non-idling and has a threshold-type structure. Our results extend previous work on systems with additive service rates. When the aforementioned conditions are not satisfied we show by examples that the optimal policy may have counterintuitive properties, which is not the case when a fully collaborative service discipline is assumed. We also obtain novel results for any type of collaboration when idling policies may be optimal.

Modeling Curbside Parking as a Network of Finite Capacity Queues

Paid curbside parking can be advantageously modeled as a network of interdependent queues. To this end, we introduce methods for analyzing a special class of networks of finite capacity queues where drivers arrive from an exogenous source, join the queue if there is an available parking space, or continue to search at an adjacent queue for an available space. Furthermore, we apply this model to estimate the proportion of drivers cruising in the neighborhood of Belltown, Seattle, WA, USA. Using occupancy approximated by parking transaction data, we estimate the percentage of drivers cruising for curbside parking by comparing the rate of drivers unable to find parking to bulk through-traffic measurement data. We find percentages of up to 50% for a Belltown's 1st Ave. depending on the time, day, and direction of travel. We then calculate a per vehicle travel-time cost to social welfare incurred by this proportion: upward of a 10% increase in travel time to all drivers along 1st Ave. Last, we introduce a simulation tool and test assumptions made when estimating interesting performance metrics like the probability of a block-face being full. Our results suggest that while assuming exponential service time distributions is not justified, mean rate solutions under a Markovian relaxation of the problem is comparable to service times representative of parking transaction data in simulation.

IFQP-EPS: Analyzing effects of queuing and storage issues on electric power systems under dual uncertainties

In this study, an interval fuzzy queue-storage (IFQS) method is developed through integrating interval-parameter programming, fuzzy queue and storage theory into a general framework. IFQS is not only capable of reflecting the effects of queue and storage on electric power systems (EPS), but also helpful for dealing with dual uncertainties expressed as fuzzy sets and interval values. Then, an IFQS-EPS model is formulated for the City of Shijiazhuang (the capital of Hebei province), where M/M/1 (exponential interarrival time, exponential service time and one server) queue with fuzzy information and different storage periods are measured. Results reveal that (i) coal mines in Yulin, Datong and Jincheng would be the major coal suppliers for the study city (providing over 77% of coal); (ii) as waiting time adds 1 day, system cost would increase around $200 million and nitrogen dioxide (NOx) emission would grow approximately 91 ton; (iii) system cost and NOx emission would respectively decrease $370 × 106 and 90 × 103 ton when storage period varies from 10 day to 20 day; (iv) electricity generated by coal would reduce 5% due to the limitation of coal supply and the control of air pollution. Compared to interval-fuzzy queue (IFQ) approach, the developed IFQS method has advantages in balancing coal purchase batch and storage cost, such that an optimal system cost can be obtained. Results can not only facilitate to generate appropriate coal purchase and electricity generation plans, but also reveal effects of queuing and storage issues on EPS under complicated uncertainties.

The Age of Information: Real-Time Status Updating by Multiple Sources

We examine multiple independent sources providing status updates to a monitor through simple queues. We formulate an age of information (AoI) timeliness metric and derive a general result for the AoI that is applicable to a wide variety of multiple source service systems. For first-come first-served and two types of last-come first-served systems with Poisson arrivals and exponential service times, we find the region of feasible average status ages for multiple updating sources. We then use these results to characterize how a service facility can be shared among multiple updating sources. A new simplified technique for evaluating the AoI in finite-state continuous-time queuing systems is also derived. Based on stochastic hybrid systems, this method makes AoI evaluation to be comparable in complexity to finding the stationary distribution of a finite-state Markov chain.

Determining Trunk Lines in Call Centers with Nonstationary Arrivals and Lognormal Service Times

Two important resources in a call center are the number of staff and the number of trunk lines required. In this paper, we focus on the decision of the number of trunk lines that a call center should have. The current practice is to use the Erlang B or the <i>M/M/s/0</i> queueing model which assumes Poisson arrivals, exponential service times, <i>s</i> servers and no places in queue, i.e. no customers can wait. In this paper, we improve on the state of practice in determining the required number of trunk lines, by including two realistic features present in call centers. The first realistic feature is to consider nonstationarity of arrivals. The second feature is to consider the lognormal service time distribution instead of the exponential distribution. There is extensive empirical evidence for both features. In order to carry out our computations we use the results of a paper by Massey and Whitt, Operations Research, 44(6), 1996. We have two main findings. Firstly, we find numerically that in our nonstationary Erlang loss model, <i>M_t/G/s/0</i>, an insensitivity result holds. The blocking probability of arrivals at the call center depends only on the mean of the lognormal service time distribution and not on its variance. Our second finding is that current practice is quite robust. In particular, we find the number of trunk lines required using a stationary Poisson approximation. This approximation assumes stationary Poisson arrivals with an appropriately chosen arrival rate and exponential service times. The approximation does quite well in predicting the number of trunk lines required.

Optimal routeing in two-queue polling systems

AbstractWe consider a polling system with two queues, exhaustive service, no switchover times, and exponential service times with rate µ in each queue. The waiting cost depends on the position of the queue relative to the server: it costs a customercper time unit to wait in the busy queue (where the server is) anddper time unit in the idle queue (where there is no server). Customers arrive according to a Poisson process with rate λ. We study the control problem of how arrivals should be routed to the two queues in order to minimize the expected waiting costs and characterize individually and socially optimal routeing policies under three scenarios of available information at decision epochs: no, partial, and complete information. In the complete information case, we develop a new iterative algorithm to determine individually optimal policies (which are symmetric Nash equilibria), and show that such policies can be described by a switching curve. We use Markov decision processes to compute the socially optimal policies. We observe numerically that the socially optimal policy is well approximated by a linear switching curve. We prove that the control policy described by this linear switching curve is indeed optimal for the fluid version of the two-queue polling system.

Analysis and evaluation of the functionality of the mass service system on the basis of customs of truck vehicles

Paper discussed the modeling of customs processes for truck vehicles using the Markov processes and mass service theory (queue theory), showing the operation of the notification handling system as a system dependent on random events. The system is characterized as a system with Poisson input stream, exponential service time and many service stations. The results are presented in the form of graphs based on real data received from the customs office

The “Sensitive” Markovian queueing system and its application for a call center problem

In this paper, we study the \(M_n/M_n/c/K+M_n\) queueing system where customers arrive according to a Poisson process with state-dependent rates. Moreover, the rates of the exponential service times and times to abandonment of the queued customers can also change whenever the system size changes. This implies that a customer may experience different service rates throughout the time she is being served. Similarly, a queued customer can change her patience time limits while waiting in the queue. Thus, we refer to the analyzed system as the “sensitive” Markovian queue. We conduct an exact analysis of this system and obtain its steady-state performance measures. The steady-state system size distribution yields itself via a birth–death process. The times spent in the queue by an arbitrary or an eventually served customer are represented as the times until absorption in two continuous-time Markov chains and follow Phase-type distributions with which the queueing time distributions and moments are obtained. Then, we demonstrate how the \(M_n/M_n/c/K+M_n\) queue can be employed to approximately yet accurately estimate the performance measures of the \(M_n/GI/c/K+GI\) type call center.

The waiting time distribution for a correlated queue with exponential interarrival and service times

We are concerned with the analysis of the waiting time distribution in an M∕M∕1 queue in which the interarrival time between the nth and the (n+1)th customers and the service time of the nth customer are correlated random variables with Downton’s bivariate exponential distribution. In this paper we show that the conditional waiting time distribution, given that the waiting time is positive, is exponential.