COM 426 SIMULATION & MODELING
COM 426 SIMULATION & MODELING Chapter 5 Queuing Systems Introduction 2 The basic phenomenon of queuing arises whenever a shared facility needs to be accessed for service by a large number of jobs or customers. The primary tool for studying these problems [of congestions] is known as queuing theory. The study of the phenomena of standing, waiting, and serving, and we call this study Queuing Theory. Any system in which arrivals place demands upon a finite capacity resource may be termed a queuing system. COM 426 Simulation & Modeling March 17, 2015 Applications • • • • • Telecommunications Traffic control Determining the sequence of computer operations Predicting computer performance Health services (eg. control of hospital bed assignments) • Airport traffic, airline ticket sales • Layout of manufacturing systems. 3 COM 426 Simulation & Modeling March 17, 2015 The Queuing Model Queuing System Queue Server • Use Queuing models to – Describe the behavior of queuing systems – Evaluate system performance • A Queue System is characterized by – Queue (Buffer): with a finite or infinite size • The state of the system is described by the Queue Size 4 – Server: with a given processing speed COM 426 Simulation & Modeling March 17, 2015 – Events: Arrival (birth) or Departure (death) with given rates Characteristics 5 Arrival Process The distribution that determines how the tasks arrives in the system. Service Process The distribution that determines the task processing time Number of Servers Total number of servers available to process the tasks COM 426 Simulation & Modeling March 17, 2015 arrival process: 6 how customers arrive e.g. singly or in groups (batch or bulk arrivals) how the arrivals are distributed in time (e.g. what is the probability distribution of time between successive arrivals (the interarrival time distribution)) whether there is a finite population of customers or (effectively) an infinite number The simplest arrival process is one where we have completely regular arrivals (i.e. the same constant time interval between successive arrivals). COM 426 Simulation & Modeling March 17, 2015 7 A Poisson stream of arrivals corresponds to arrivals at random. In a Poisson stream successive customers arrive after intervals which independently are exponentially distributed. The Poisson stream is important as it is a convenient mathematical model of many real life queuing systems and is described by a single parameter the average arrival rate. Other important arrival processes are scheduled arrivals; batch arrivals; and time dependent arrival rates (i.e. the arrival rate varies according to the time of day). COM 426 Simulation & Modeling March 17, 2015 service mechanism: 8 a description of the resources needed for service to begin how long the service will take (the service time distribution) the number of servers available whether the servers are in series (each server has a separate queue) or in parallel (one queue for all servers) whether preemption is allowed (a server can stop processing a customer to deal with another "emergency" customer) First-come-first-served(FCFS) Last-come-first-served(LCFS) Shortest processing time first(SPT) Shortest remaining processing time first(SRPT) Shortest expected processing time first(SEPT) Shortest expected remaining processing time first(SERPT) Biggest-in-first-served(BIFS) Loudest-voice-first-served(LVFS) COM 426 Simulation & Modeling March 17, 2015 Queue characteristics: 9 how, from the set of customers waiting for service, do we choose the one to be served next (e.g. FIFO (first-in firstout) - also known as FCFS (first-come first served); LIFO (last-in first-out); randomly) (this is often called the queue discipline) do we have: balking (customers deciding not to join the queue if it is too long) reneging (customers leave the queue if they have waited too long for service) jockeying (customers switch between queues if they think they will get served faster by so doing) a queue of finite capacity or (effectively) of infinite capacity COM 426 Simulation & Modeling March 17, 2015 Typical Distributions 10 M :Markovian (Exponential / poisson) Ek : Erlang with parameter k Hk : Hyperexponential with parameter k(mixture of k exponentials) D : Deterministic(constant) G : General(all) COM 426 Simulation & Modeling March 17, 2015 Model Notations 11 Kendall’s Notation: A/S/m/B/K/SD A: arrival process S: service time distribution m: number of servers B: number of buffers(system capacity) K: population size SD: service discipline COM 426 Simulation & Modeling March 17, 2015 12 Standard: A/B/C/D/E,F A represents the probability distribution for the arrival process B represents the probability distribution for the service process C represents the number of channels (servers) D represents the maximum number of customers allowed in the queuing system (either being served or waiting for service) E represents the maximum number of customers in total F represents the queue discipline COM 426 Simulation & Modeling March 17, 2015 Examples 13 M/M/3/20/1500/FCFS Time between successive arrivals is exponentially distributed Service times are exponentially distributed Three servers 20 buffers = 3 service + 17 waiting After 20, all arriving jobs are lost Total of 1500 jobs that can be serviced Service discipline is first-come-first-served COM 426 Simulation & Modeling March 17, 2015 14 MM1 The most commonly used type of queue Used to model single processor systems or individual devices in a computer system Assumption rate of exponentially distributed Service rate of exponentially distributed Single server FCFS Unlimited queue lengths allowed Infinite number of customers Interarrival COM 426 Simulation & Modeling March 17, 2015 An M/M/1 Queueing Example = mean number of arrivals per time period = mean number of people or items served per time period Average number of customers in the system LS = Average time a customer spends in the system 1 WS = Average number of customers waiting in the queue Lq = 2 Average time a customer spends waiting in the queue Wq = Utilization factor for the system = 15 COM 426 Simulation & Modeling March 17, 2015 Probability of 0 customers in the system P0 = 1 - Probability of more than k customers in the system Pk = k 1 Example 1: Peter’s Drive-In on 16th Avenue has one walk-up service window for people who want to park their cars and eat at a picnic table. On the Friday before a long weekend, customers arrive at the window at a rate of 30 per hour, following a Poisson distribution. There is a very large, almost infinite number of hungry customers at this time of the year. The customers are served on a first come, first served basis, and it takes approximately 1.5 minutes to serve each customer. 16 COM 426 Simulation & Modeling March 17, 2015 Example 2 17 You are entering Bank of America Arena at Hec Edmunson Pavilion to watch a basketball game. There is only one ticket line to purchase tickets. Each ticket purchase takes an average of 18 seconds. The average arrival rate is 3 persons/minute. Find the average length of queue and average waiting time in queue assuming M/M/1 queuing. COM 426 Simulation & Modeling March 17, 2015
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