Introduction to Queuing and Simulation.ppt
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1、1,Introduction to Queuing and Simulation,Chapter 6Business Process Modeling, Simulation and Design,2,Overview (I),What is queuing/ queuing theory? Why is it an important tool? Examples of different queuing systems Components of a queuing system The exponential distribution & queuing Stochastic proce
2、sses Some definitions The Poisson process Terminology and notation Littles formula Birth and Death Processes,3,Overview (II),Important queuing models with FIFO discipline The M/M/1 model The M/M/c model The M/M/c/K model (limited queuing capacity) The M/M/c/N model (limited calling population) Prior
3、ity-discipline queuing models Application of Queuing Theory to system design and decision making,4,Overview (III),Simulation What is that? Why is it an important tool? Building a simulation model Discrete event simulation Structure of a BPD simulation project Model verification and validation Exampl
4、e Simulation of a M/M/1 Queue,5,Mathematical analysis of queues and waiting times in stochastic systems. Used extensively to analyze production and service processes exhibiting random variability in market demand (arrival times) and service times. Queues arise when the short term demand for service
5、exceeds the capacity Most often caused by random variation in service times and the times between customer arrivals. If long term demand for service capacity the queue will explode!,What is Queuing Theory?,6,Capacity problems are very common in industry and one of the main drivers of process redesig
6、n Need to balance the cost of increased capacity against the gains of increased productivity and service Queuing and waiting time analysis is particularly important in service systems Large costs of waiting and of lost sales due to waitingPrototype Example ER at County Hospital Patients arrive by am
7、bulance or by their own accord One doctor is always on duty More and more patients seeks help longer waiting times Question: Should another MD position be instated?,Why is Queuing Analysis Important?,7,A Cost/Capacity Tradeoff Model,8,Commercial Queuing Systems Commercial organizations serving exter
8、nal customers Ex. Dentist, bank, ATM, gas stations, plumber, garage Transportation service systems Vehicles are customers or servers Ex. Vehicles waiting at toll stations and traffic lights, trucks or ships waiting to be loaded, taxi cabs, fire engines, elevators, buses Business-internal service sys
9、tems Customers receiving service are internal to the organization providing the service Ex. Inspection stations, conveyor belts, computer support Social service systems Ex. Judicial process, the ER at a hospital, waiting lists for organ transplants or student dorm rooms ,Examples of Real World Queui
10、ng Systems?,9,Components of a Basic Queuing Process,Calling Population,Queue,Service Mechanism,Input Source,The Queuing System,Jobs,Arrival Process,Queue Configuration,Queue Discipline,Served Jobs,Service Process,leave the system,10,The calling population The population from which customers/jobs ori
11、ginate The size can be finite or infinite (the latter is most common) Can be homogeneous (only one type of customers/ jobs) or heterogeneous (several different kinds of customers/jobs) The Arrival Process Determines how, when and where customer/jobs arrive to the system Important characteristic is t
12、he customers/jobs inter-arrival times To correctly specify the arrival process requires data collection of interarrival times and statistical analysis.,Components of a Basic Queuing Process (II),11,The queue configuration Specifies the number of queues Single or multiple lines to a number of service
13、 stations Their location Their effect on customer behavior Balking and reneging Their maximum size (# of jobs the queue can hold) Distinction between infinite and finite capacity,Components of a Basic Queuing Process (III),12,Example Two Queue Configurations,13,The service provided can be differenti
14、ated Ex. Supermarket express lanes Labor specialization possible Customer has more flexibility Balking behavior may be deterred Several medium-length lines are less intimidating than one very long line,Guarantees fairness FIFO applied to all arrivals No customer anxiety regarding choice of queue Avo
15、ids “cutting in” problems The most efficient set up for minimizing time in the queue Jockeying (line switching) is avoided,Multiple v.s. Single Customer Queue Configuration,Multiple Line Advantages,Single Line Advantages,14,The Service Mechanism Can involve one or several service facilities with one
16、 or several parallel service channels (servers) - Specification is required The service provided by a server is characterized by its service time Specification is required and typically involves data gathering and statistical analysis. Most analytical queuing models are based on the assumption of ex
17、ponentially distributed service times, with some generalizations. The queue discipline Specifies the order by which jobs in the queue are being served. Most commonly used principle is FIFO. Other rules are, for example, LIFO, SPT, EDD Can entail prioritization based on customer type.,Components of a
18、 Basic Queuing Process (IV),15,Concealing the queue from arriving customers Ex. Restaurants divert people to the bar or use pagers, amusement parks require people to buy tickets outside the park, banks broadcast news on TV at various stations along the queue, casinos snake night club queues through
19、slot machine areas. Use the customer as a resource Ex. Patient filling out medical history form while waiting for physician Making the customers wait comfortable and distracting their attention Ex. Complementary drinks at restaurants, computer games, internet stations, food courts, shops, etc. at ai
20、rports Explain reason for the wait Provide pessimistic estimates of the remaining wait time Wait seems shorter if a time estimate is given. Be fair and open about the queuing disciplines used,Mitigating Effects of Long Queues,16,A Commonly Seen Queuing Model (I),C C C C,Customers (C),C S = Server C
21、S C S,Customer =C,The Queuing System,The Queue,The Service Facility,17,Service times as well as interarrival times are assumed independent and identically distributed If not otherwise specified Commonly used notation principle: A/B/C A = The interarrival time distribution B = The service time distri
22、bution C = The number of parallel servers Commonly used distributions M = Markovian (exponential) - Memoryless D = Deterministic distribution G = General distribution Example: M/M/c Queuing system with exponentially distributed service and inter-arrival times and c servers,A Commonly Seen Queuing Mo
23、del (II),18,The most commonly used queuing models are based on the assumption of exponentially distributed service times and interarrival times.Definition: A stochastic (or random) variable Texp( ), i.e., is exponentially distributed with parameter , if its frequency function is:,The Exponential Dis
24、tribution and Queuing, The Cumulative Distribution Function is: The mean = ET = 1/ The Variance = VarT = 1/ 2,19,The Exponential Distribution,Time between arrivals,Mean= ET=1/,Probability density,t,fT(t),20,Property 1: fT(t) is strictly decreasing in t P(0Tt) P(t T t+t) for all t, t0Implications Man
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