Simple Queueing Theory: Page 5.1 CPE Systems Modelling & Simulation Techniques Topic 5: Simple Queueing Theory  Queueing Models  Kendall notation  Steady state analysis  Performance measures  Different queue models

Simple Queueing Theory: Page 5.2 CPE 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

Simple Queueing Theory: Page 5.3 CPE 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

Simple Queueing Theory: Page 5.4 CPE 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.

Simple Queueing Theory: Page 5.5 CPE 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

Simple Queueing Theory: Page 5.6 CPE 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.

Simple Queueing Theory: Page 5.7 CPE 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.

Simple Queueing Theory: Page 5.8 CPE 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

Simple Queueing Theory: Page 5.9 CPE 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).

Simple Queueing Theory: Page 5.10 CPE 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

Simple Queueing Theory: Page 5.11 CPE Systems Modelling & Simulation Techniques M/M/1 example

Simple Queueing Theory: Page 5.12 CPE 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.

Simple Queueing Theory: Page 5.13 CPE Systems Modelling & Simulation Techniques M/M/1/N/  example

Simple Queueing Theory: Page 5.14 CPE 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.

Simple Queueing Theory: Page 5.15 CPE Systems Modelling & Simulation Techniques M/M/c example

Simple Queueing Theory: Page 5.16 CPE 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.