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Simple Queueing Theory: Page 5.1 CPE1005 - Systems Modelling & Simulation Techniques Topic 5: Simple Queueing Theory Queueing Models Kendall notation Steady state analysis Performance measures Different queue models
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Simple Queueing Theory: Page 5.2 CPE1005 - Systems Modelling & Simulation Techniques Queues and components Queues are frequently used in simulations. Population: The entity (“customers”) that requires service Server: The entity that provides the service Queue: The entity that tempoparily holds the waiting “customers” before they are served. Events: arrival, service, and leaving. Calling Population Waiting line Server arrival leaving service
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Simple Queueing Theory: Page 5.3 CPE1005 - Systems Modelling & Simulation Techniques Purpose of Queueing Models Most models are to determine the level of service Two major factors: Cost of providing service: cannot afford many idle servers. Cost of customer dissatisfaction: customers will leave if queue is too long. Tradeoff between these 2 factors Service level Cost Total cost Cost of providing service Cost of customer dissatisfaction
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Simple Queueing Theory: Page 5.4 CPE1005 - Systems Modelling & Simulation Techniques Characteristics of Queue models Calling population infinite population: leads to simpler model, useful when number of potential “customers” >> number of “customers” in system. Finite population: arrival rate is affected by the number of customers already in the system. System capacity The number of customers that can be in the queue or under service. An infinite capacity means no customer will exit prematurely. Arrival process For infinite population, arrival process is defined by the interarrival times of successive customers Arrivals can be scheduled or at random times, Poisson dist’n is used frequently for random arrivals, and scheduled arrivals usually use a constant interarrival rate.
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Simple Queueing Theory: Page 5.5 CPE1005 - Systems Modelling & Simulation Techniques Characteristics of Queue models Queue behaviour describes how the customer behaves while in the queue waiting balking - leave when they see the line is too long renege - leave after being in the queue for too long jockey - move from one queue to another Queue discipline FIFO - first in first out (most common) FILO - first in last out (stack) SIRO - service in random order SPT - shortest processing time first PR - service based on priority
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Simple Queueing Theory: Page 5.6 CPE1005 - Systems Modelling & Simulation Techniques Characteristics of Queue models Service Times random: mainly modeled by using exponential distribution or truncated normal distribution (truncate at 0). Constant Service mechanism describes how the servers are configured. Parallel - multiple servers are operating and take customer in from the same queue. Serial - customers have to go through a series of servers before completion of service combinations of parallel and serial.
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Simple Queueing Theory: Page 5.7 CPE1005 - Systems Modelling & Simulation Techniques Kendall Notations Kendall defined the notations for parallel server systems A / B / c / N / K A: interarrival distribution type B: service time distribution type Common symbols for A, B are M for exponential, D for constant, E k for Erlang, G for general or arbitary. c: for number of parallel servers N: for system capacity K: for size of calling population.
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Simple Queueing Theory: Page 5.8 CPE1005 - Systems Modelling & Simulation Techniques Queue Characteristics and Metrics Characteristics customer arrival rate (in customers per time unit) service rate of one server (in service/transaction per time unit) Performance metrics average utilisation factor, percentage the server is busy. L q average length of queue Laverage number of customers in the system W q average waiting time in queue Waverage time spent in the system P n Probability of n customers in the system
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Simple Queueing Theory: Page 5.9 CPE1005 - Systems Modelling & Simulation Techniques Transient and long term behaviour Queue metrics changes whenever state change events happen, e.g. customers in queue at time t, service time for customer n, etc. Average metrics such as average customers in system L, average utilisation factor will vary but will approach a steady state or long term value. For simple queues, the long term metrics can be calculated analytically, based on the queue characteristics (, for M/M queues) and the initial conditions (whether a customer is already under service, whether customers are already queuing at time 0).
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Simple Queueing Theory: Page 5.10 CPE1005 - Systems Modelling & Simulation Techniques M/M/1/ / or M/M/1 model One of the basic queueing models. Single server, both arrival rate and service rate are exponential
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Simple Queueing Theory: Page 5.11 CPE1005 - Systems Modelling & Simulation Techniques M/M/1 example
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Simple Queueing Theory: Page 5.12 CPE1005 - Systems Modelling & Simulation Techniques M/M/1/N/ Single server queue, fixed length Fixed length queue means customer will not get into the system if the maximum system capacity is filled up.
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Simple Queueing Theory: Page 5.13 CPE1005 - Systems Modelling & Simulation Techniques M/M/1/N/ example
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Simple Queueing Theory: Page 5.14 CPE1005 - Systems Modelling & Simulation Techniques Adding more servers M/M/c Complicated formula to find P 0, probability that all servers are empty, and P , probability that all servers are busy.
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Simple Queueing Theory: Page 5.15 CPE1005 - Systems Modelling & Simulation Techniques M/M/c example
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Simple Queueing Theory: Page 5.16 CPE1005 - Systems Modelling & Simulation Techniques Other models M/M/c/K/K This is used to model a finite number of calling population. E.g. a restaurant with X tables of customers and Y waiters to serve the customers. M/D/1 Service time has no variation. D/M/1 deterministic arrival pattern, with exponential service time. E.g. a doctor’s timetable with appointments. M/E k /1 Service follows an Erlang distribution. E.g. a series of procedures that take the same average time to complete for each.
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