1 / 16

Topic 5: Simple Queueing Theory

Topic 5: Simple Queueing Theory. Queueing Models Kendall notation Steady state analysis Performance measures Different queue models. Queues and components. Queues are frequently used in simulations. Population: The entity (“customers”) that requires service

niveditha
Download Presentation

Topic 5: Simple Queueing Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Topic 5: Simple Queueing Theory • Queueing Models • Kendall notation • Steady state analysis • Performance measures • Different queue models

  2. 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. leaving Calling Population service arrival Server Waiting line

  3. 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 Cost Total cost Cost of providing service Cost of customer dissatisfaction Service level

  4. 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.

  5. 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

  6. 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.

  7. 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, Ek for Erlang, G for general or arbitary. • c: for number of parallel servers • N: for system capacity • K: for size of calling population.

  8. 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. • Lq average length of queue • L average number of customers in the system • Wq average waiting time in queue • W average time spent in the system • Pn Probability of n customers in the system

  9. 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).

  10. M/M/1// or M/M/1 model • One of the basic queueing models. • Single server, both arrival rate  and service rate  are exponential

  11. M/M/1 example

  12. 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.

  13. M/M/1/N/ example

  14. Adding more servers M/M/c • Complicated formula to find P0, probability that all servers are empty, and P, probability that all servers are busy.

  15. M/M/c example

  16. 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/Ek/1 • Service follows an Erlang distribution. E.g. a series of procedures that take the same average time to complete for each.

More Related