A two-class global FCFS discrete-time queueing model with arbitrary-length constant service times (original) (raw)
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Performance Evaluation, 2017
In this paper, we analyze a discrete-time queueing model with two types (classes) of customers and two servers, one for each customer class. Although each server can only process one type of customers, all customers are accommodated in one single queue and served in their order of arrival, irrespective of their types. The numbers of customers arriving in the system from time slot to time slot are independent, but the types of consecutive customers are not necessarily independent. Specifically, we assume that a first-order Markovian correlation ("interclass correlation") exists between the types of subsequent customers in the arrival stream. The fact that multiple customers of the same type may arrive back-to-back and customers have to be served in their order of arrival, causes occasional under-utilization of the service capacity of the system, because some customers may not be able to reach their server owing to the presence of customers of the opposite type in front of them. In this paper, we assume that the service times of both types of customers are independent, geometrically distributed random variables. The paper extends earlier work where all the service times were assumed to be of fixed length, either equal to 1 slot each, or equal to multiple slots. The fact that, in the present paper, service times are of variable length, entails that customers being served simultaneously can overtake each other, thus disturbing the original arrival order. This phenomenon did not occur in previous studies with fixed-length service times, and represents the main new element of the paper. It also complicates the analysis of the system considerably. Nevertheless, we are able to derive explicit expressions for the probability generating functions and the mean values of the main performance measures of the system, in terms of the original system parameters and one root of a non-linear equation. Our results reveal the impact of the interclass correlation and the variable nature of the service times on the achievable throughput, the (mean) number of customers in the system, the (mean) customer sojourn times, the (mean) unfinished work in the system, and related quantities.
A continuous-time queueing model with class clustering and global FCFS service discipline
Journal of Industrial and Management Optimization, 2013
ABSTRACT In this paper the focus is on “class clustering” in a continuous-time queueing model with two classes and dedicated servers. “Class clustering” means that customers of any given type may (or may not) have a tendency to “arrive back-to-back”. We believe this is a concept that is often neglected in literature and we want to show that it can have a considerable impact on multiclass queueing systems, especially on the system considered in this paper. This system adopts a “global FCFS” service discipline, i.e., all arriving customers are accommodated in one single FCFS queue, regardless of their types. The major aim of our paper is to quantify the intuitively expected (due to the service discipline) negative impact of “class clustering” on the performance measures of our system. The motivation of our work are systems where this kind of inherent blocking is encountered, such as input-queueing network switches, road splits or security checks at airports.
Analysis of a two-class single-server discrete-time FCFS queue: the effect of interclass correlation
TOP
In this paper we study a discrete-time queueing system with one server and two classes of customers. Customers enter the system according to a general independent arrival process. The classes of consecutive customers, however, are correlated in a Markovian way. The system uses a "global FCFS" service discipline, i.e., all arriving customers are accommodated in one single FCFS queue, regardless of their classes. The service-time distribution of the customers is general but class-dependent, and therefore, the exact order in which the customers of both classes succeed each other in the arrival stream is important, which is reflected by the complexity of the system content and waiting time analysis presented in this paper. In particular, a detailed waiting time analysis of this kind of multiclass system has not yet been published, and is considered to be one of the main novelties by the authors. In addition to that, a major aim of the paper is to estimate the impact of interclass correlation in the arrival stream on the total number of customers in the system, and the customer delay. The results reveal that the system can exhibit two different classes of stochastic equilibrium: a "strong" equilibrium where both customer classes give rise to stable behavior individually, and a "compensated" equilibrium where one customer type creates overload.
The impact of a global FCFS service discipline in a two-class queue with dedicated servers
Computers & Operations Research, 2016
This article discusses the steady analysis of a discrete time queue of Geo/Geo+G/2 type. All arriving customers are served either by server-1 according to a geometrically distributed service time S 1 =k slots for k=1,2, … ∞ , with mass function f 1 (k)==Pr(S 1 =k) = μ(1-μ) k-1 with mean rate 0<μ<1 or by server-2 with a general service time S 2 = k for k=1,2, … ∞ , with mass function f 2 (k)==Pr(S 2 =k) with mean service time is β= 2 1 () k k f k ∞ = ∑ or mean service rate μ 2 =1/β. Sequel to some objections raised on the use of the classical 'First Come First Served (FCFS)' queue discipline when the two heterogeneous servers operate as parallel service providers, an alternative queue discipline in a serial configuration of servers are considered in this work; the objective is that if, in a singlechannel queue in equilibrium, the service rate suddenly increases and exceeds the present service capacity, install a new channel to work serially with the first channel as suggested by Krishnamoorthy (1968). Using the embedded method subject to different service time distributions we present an exact analysis for finding the 'Probability generating Function (PGF)' of steady state number of customers in the system and most importantly, the actual waiting time expectation of customers in the system. This work shows that one can obtain all stationery probabilities and other vital measures for this queue under certain additional and simple but realistic assumptions.
Analysis Of A FCFS Queue With Two Types Of Customers And Order-Dependent Service Times
ECMS 2014 Proceedings edited by: Flaminio Squazzoni, Fabio Baronio, Claudia Archetti, Marco Castellani, 2014
In this paper, we study a discrete-time first-comefirst-served queueing system with a single server and two types (classes) of customers, where the (average) service time of a customer is longer if its type differs from the type of the preceding customer. As opposed to traditional literature, the different types of customers do not occur randomly and independently in the arrival stream: we include a Markovian type of correlation in the types of consecutive customers instead. We deduce the probability generating function of the system content, from which we extract various performance measures, such as the mean values of the system content and the customer delay. We demonstrate that the interclass correlation in the arrival stream has a tremendous impact on the system performance, which highlights the necessity to include it in the performance assessment of the system.
Impact of class clustering in a multiclass FCFS queue with order-dependent service times
Computers & Operations Research, 2014
In multi-class queueing systems, customers of different classes can enter the system. When studying such systems, it is traditionally assumed that the different classes of customers occur randomly and independently in the arrival stream of customers in the system. This is often in contrast to the actual situation. Therefore, we study a multi-class system with so-called class clustering in the customer arrival stream, i.e., (Markovian) correlation occurs in the classes of consecutive customers. The system under investigation consists of one server that is able to serve two classes of customers. In addition, the service-time distribution of a customer depends on the equality or non-equality of its class with the class of the previous customer. This latter feature occurs frequently in practice. For instance, execution of the same task again can lead to both faster or slower processing times. The first case can occur when the execution of a different task entails resetting a machine, or loading new data, et cetera. The opposite situation appears, for instance, when execution of the same task requires postprocessing (such as cooling down or reinitialisation of a machine). We deduce the probability generating function (pgf) of the system content, from which we can extract various performance measures, among which the mean values of the system content and the customer delay. We demonstrate that class clustering has a tremendous impact on the system performance, which highlights the necessity to include it in the performance assessment of any system in which it occurs.
A multi-class discrete-time queueing system under the FCFS service discipline
Annals of Operations Research, 2012
The problem with the FCFS server discipline in discrete-time queueing systems is that it doesn't actually determine what happens if multiple customers enter the system at the same time, which in the discrete-time paradigm translates into 'during the same timeslot'. In other words, it doesn't specify in which order such customers are served. When we consider multiple types of customers, each requiring different service time distributions, the precise order of service even starts to affect quantities such as queue content and delays of arbitrary customers, so specifying this order will be prime. In this paper we study a multiclass discrete-time queueing system with a general independent arrival process and generally distributed service times. The service discipline is FCFS and customers entering during the same time-slot are served in random order. It will be our goal to search for the steady-state distribution of queue content and delays of certain types of customers. If one thinks of the time-slot as a continuous but bounded time period, the random order of service is equivalent to FCFS if different customers have different arrival epochs within this time-slot and if the arrival epochs are independent of customer class. For this reason we propose two distinct ways of analysing; one utilizing permutations, the other considering a slot as a bounded continuous time frame.
Analysis of a Two-Class FCFS Queueing System with Interclass Correlation
This paper considers a discrete-time queueing system with one server and two classes of customers. All arriving customers are accommodated in one queue, and are served in a First-Come-First-Served order, regardless of their classes. The total numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution. The classes of consecutively arriving customers, however, are correlated in a Markovian way, i.e., the probability that a customer belongs to a class depends on the class of the previously arrived customer. Service-time distributions are assumed to be general but class-dependent. We use probability generating functions to study the system analytically. The major aim of the paper is to estimate the impact of the interclass correlation in the arrival stream on the queueing performance of the system, in terms of the (average) number of customers in the system and the (average) customer delay and customer waiting time.
Heavy Traffic Analysis of Multi-Class Bipartite Queueing Systems Under FCFS
SSRN Electronic Journal
This paper examines the performance of multi-class multi-server bipartite queueing systems under a FCFS-ALIS service discipline, where each arriving customer is only compatible with a subset of servers. We analyze the system under conventional heavy-traffic conditions, where the traffic intensity approaches one from below. Building upon the formulation and results of Afèche et al. (2022), we generalize the model by allowing the vector of arrival rates to approach the heavytraffic limit from an arbitrary direction. We characterize the steady-state waiting times of the various customer classes and demonstrate that a much wider range of waiting time outcomes is achievable. Furthermore, we establish that the matching probabilities, i.e., the probabilities of different customer classes being served by different servers, do not depend on the direction along which the system approaches heavy traffic. We also investigate the design of compatibility between customer classes and servers, finding that a service provider who has complete control over the matching can design a delay-minimizing menu by considering only the limiting arrival rates. When some constraints on the compatibility structure exist, the direction of convergence to heavy-traffic affects which menu minimizes delay. Additionally, we discover that the bipartite matching queueing system exhibits a form of Braess's paradox, where adding more connectivity to an existing system can lead to higher average waiting times, despite the fact that neither customers nor servers act strategically.
Discrete-time queues with variable service capacity: a basic model and its analysis
Annals of Operations Research, 2013
In this paper, we present a basic discrete-time queueing model whereby the service process is decomposed in two (variable) components: the demand of each customer, expressed in a number of work units needed to provide full service of the customer, and the capacity of the server, i.e., the number of work units that the service facility is able to perform per time unit. The model is closely related to multi-server queueing models with server interruptions, in the sense that the service facility is able to deliver more than one unit of work per time unit, and that the number of work units that can be executed per time unit is not constant over time. Although multi-server queueing models with server interruptions-to some extentallow us to study the concept of variable capacity, these models have a major disadvantage. The models are notoriously hard to analyze and even when explicit expressions are obtained, these contain various unknown probabilities that have to be calculated numerically, which makes the expressions difficult to interpret. For the model in this paper, on the other hand, we are able to obtain explicit closed-form expressions for the main performance measures of interest. Possible applications of this type of queueing model are numerous: the variable service capacity could model variable available bandwidths in communication networks, a varying production capacity in factories, a variable number of workers in an HR-environment, varying capacity in road traffic, etc.