Topic 5: Simple Queueing Theory

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.