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Operations Research
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Paul L. Goethals, Natalie M. Scala, Nathaniel D. Bastian
Queueing system structures can be represented in various ways and are usually characterized by a finite or infinite-capacity queue, some type of service policy, and some arrangement of one or more servers. When describing various queueing systems frequently observed in research, Kendall notation is commonly used, which was first suggested by Kendall (1953). The long variation of this notation consists of a partition ‘(i) / (ii) / (iii) / (iv) / (v) / (vi)’ whereby each element represents (i) the distribution of arrival times, (ii) the distribution of service times, (iii) the number of servers, (iv) the capacity of the queue, (v) the population size, and (vi) the queueing policy used. Most often, the assumptions of an infinite queue capacity, infinite population size, and a first-in first-out (FIFO) queueing policy reduces the six-element long notation to an abbreviated three-element variation.
Basic Queueing Theory
Published in Samyukta Sethuraman, Analysis of Fork-Join Systems, 2022
The inter-arrival and service time distributions constitute the first two components of the Kendall notation. The parameters associated with these distributions have a significant impact on the performance of the queueing system. Some of the critical parameters of inter-arrival and service time distributions needed for quantifying the performance metrics of a queueing system are listed below: Mean arrival rate (λ): This parameter is the average number of jobs entering the queueing system per unit time, i.e.λ is equal to the inverse of the mean of the inter-arrival time distribution.Mean service rate (μ): This parameter is the average number of jobs whose service can be completed by one server per unit time, i.e.μ is the inverse of the mean of the service time distribution.Traffic intensity (ρ): The average load experienced by the queueing system is denoted by ρ and is related to λ and μ as: ρ=λsμ
1 Introduction
Published in Paul J. Fortier, George R. Desrochers, Modeling and Analysis of Local Area Networks, 1990
Paul J. Fortier, George R. Desrochers
The notation used (see Figure 1-2) to describe the queue phenomenon is as follows: The arrival distribution defines the arrival patterns of customers into the queue. These are defined by a random variable that defines the interarrival time. A typically used measure is the Poisson arrival process defined as : P[arrival ≤ time]=l-e−λt; where the average arrival rate is λ. The queue is defined as a storage reservoir for customers. Additionally, the policy it uses for accepting and removing customers is also defined. Examples of queuing disciplines typically used are first in first out (FIFO) and last in first out (LIFO). The last main component of the queue description is the service policy, which is the method by which customers are accepted for service and the length of the service. This service time is described by a distribution, a random variable. A typical service time distribution is the random service given by Ws (t) = 1−e−μt, where t > 0, and the symbol μ is reserved to describe this common distribution for its average service rate. The distributions used to describe the arrival rate and service ratios are many and variable; for example, the exponential, general, Erlang, deterministic, or hyper-exponential can be used. The Kendall notation was developed to describe what type of queue is being examined. The form of this notation is: A/B/c/K/m/Z
Routing and staffing in emergency departments: A multiclass queueing model with workload dependent service times
Published in IISE Transactions on Healthcare Systems Engineering, 2023
Siddhartha Nambiar, Maria E. Mayorga, Yunan Liu
Our article is related to the vast literature on fluid approximations for queues. In what follows, we introduce different queueing models following the standard Kendall’s notation;2 see Ross (2019) for example. We hereby only review Many-Server Heavy-Traffic (MSHT) fluid queues that are most closely related to the present work. Heavy-traffic fluid and diffusion limits were developed by Mandelbaum et al. (1998) for time-varying Markovian queueing networks with Poisson arrivals and exponential service times. Adopting a two-parameter queue length descriptor, the pioneering work by Whitt (2006a) studied the fluid model having non-exponential service and abandonment times. Whitt (2006b) confirms that the discrete-time setting can be used as an approximation for the continuous-time setting (of the model) as time increments of the discrete-time setting can be arbitrarily short. Scaling a stochastic system to its fluid limits has been shown to be asymptotically correct in the scaled regime for the Markovian model (Mandelbaum & Pats, 1995; Whitt, 2004) and for a discrete-time analog of the general model (Whitt, 2006a). Extending the work in Whitt (2006a), Liu and Whitt (2012a) developed a fluid approximation for the queue with time-varying arrivals and non-exponential distributions; they later extended it to the framework of fluid networks (Liu and Whitt (2011), Liu and Whitt (2014a)). A functional weak law of large numbers (FWLLN) (Liu & Whitt, 2012b) was established to substantiate the fluid approximation in Liu and Whitt (2012a) and functional central limit theorems (FCLTs) were developed for the model by Liu and Whitt (2014b) and for the overloaded model by Aras et al. (2018).