IE406, I & IEUTKSimulationQueueing ModelsDr. Xueping LiUniversity of TennesseeBased onBanks, Carson, Nelson & NicolDiscrete-Event System Simulation2Purpose Simulation is often used in the analysis of queueing models. A simple but typical queueing model: Queueing models provide the analyst with a powerful tool for designing and evaluating the performance of queueing systems. Typical measures of system performance: Server utilization, length of waiting lines, and delays of customers For relatively simple systems, compute mathematically For realistic models of complex systems, simulation is usually required.3Outline Discuss some well-known models (not development of queueing theories): General characteristics of queues, Meanings and relationships of important performance measures, Estimation of mean measures of performance. Effect of varying input parameters, Mathematical solution of some basic queueing models.4Characteristics of Queueing Systems Key elements of queueing systems: Customer: refers to anything that arrives at a facility and requires service, e.g., people, machines, trucks, emails. Server: refers to any resource that provides the requested service, e.g., repairpersons, retrieval machines, runways at airport.5Calling Population[Characteristics of Queueing System] Calling population: the population of potential customers, may be assumed to be finite or infinite. Finite population model: if arrival rate depends on the number of customers being served and waiting, e.g., model of one corporatejet, if it is being repaired, the repair arrival rate becomes zero. Infinite population model: if arrival rate is not affected by the number of customers being served and waiting, e.g., systems with large population of potential customers.6System Capacity[Characteristics of Queueing System] System Capacity: a limit on the number of customers that may be in the waiting line or system. Limited capacity, e.g., an automatic car wash only has room for 10 cars to wait in line to enter the mechanism. Unlimited capacity, e.g., concert ticket sales with no limit on the number of people allowed to wait to purchase tickets.7Arrival Process[Characteristics of Queueing System] For infinite-population models: In terms of interarrival times of successive customers. Random arrivals: interarrival times usually characterized by a probability distribution. Most important model: Poisson arrival process (with rate λ), where Anrepresents the interarrival time between customer n-1 and customer n, and is exponentially distributed (with mean 1/λ). Scheduled arrivals: interarrival times can be constant or constant plus or minus a small random amount to represent early or late arrivals. e.g., patients to a physician or scheduled airline flight arrivals to an airport. At least one customer is assumed to always be present, so the server is never idle, e.g., sufficient raw material for a machine.8Arrival Process[Characteristics of Queueing System] For finite-population models: Customer is pending when the customer is outside the queueing system, e.g., machine-repair problem: a machine is “pending”when it is operating, it becomes “not pending” the instant it demands service form the repairman. Runtime of a customer is the length of time from departure from the queueing system until that customer’s next arrival to the queue, e.g., machine-repair problem, machines are customers and a runtime is time to failure. Let A1(i), A2(i), … be the successive runtimes of customer i, and S1(i), S2(i)be the corresponding successive system times:9Queue Behavior and Queue Discipline[Characteristics of Queueing System] Queue behavior: the actions of customers while in a queue waiting for service to begin, for example: Balk: leave when they see that the line is too long, Renege: leave after being in the line when its moving too slowly, Jockey: move from one line to a shorter line. Queue discipline: the logical ordering of customers in a queue that determines which customer is chosen for service when a server becomes free, for example: First-in-first-out (FIFO) Last-in-first-out (LIFO) Service in random order (SIRO) Shortest processing time first (SPT) Service according to priority (PR).10Service Times and Service Mechanism[Characteristics of Queueing System] Service times of successive arrivals are denoted by S1, S2, S3. May be constant or random. {S1, S2, S3, …} is usually characterized as a sequence of independent and identically distributed random variables, e.g., exponential, Weibull, gamma, lognormal, and truncated normal distribution. A queueing system consists of a number of service centers and interconnected queues. Each service center consists of some number of servers, c, working in parallel, upon getting to the head of the line, a customer takes the 1stavailable server.11Service Times and Service Mechanism[Characteristics of Queueing System] Example: consider a discount warehouse where customers may: Serve themselves before paying at the cashier:12Service Times and Service Mechanism[Characteristics of Queueing System] Wait for one of the three clerks: Batch service (a server serving several customers simultaneously), or customer requires several servers simultaneously.13Queueing Notation[Characteristics of Queueing System] A notation system for parallel server queues: A/B/c/N/K A represents the interarrival-time distribution, B represents the service-time distribution, c represents the number of parallel servers, N represents the system capacity, K represents the size of the calling population.14Queueing Notation[Characteristics of Queueing System] Primary performance measures of queueing systems: Pn: steady-state probability of having n customers in system, Pn(t): probability of n customers in system at time t,λ: arrival rate,λe: effective arrival rate,µ: service rate of one server,ρ: server utilization, An: interarrival time between customers n-1 and n, Sn: service time of the nth arriving customer, Wn: total time spent in system by the nth arriving customer, WnQ: total time spent in the waiting line by customer n, L(t): the number of customers in system at time t, LQ(t): the number of customers in queue at time t, L: long-run time-average number of customers in system, LQ:
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