Queuing Theory 2 HSPM J716

Simple queue model assumes … Constant average arrival rate λ and service rate μ Independence – One arrival doesn’t make another arrival more or less likely – The average length of service doesn’t change regardless of How many are waiting How busy the server has been

Greek letters ρ (“rho”) server utilization factor = average arrival rate λ ⁄ μ average service rate Probability of n in system = (1-ρ)ρ n Probability of 0 to n in system = 1-ρ n+1

Customers in System and in Queue L – mean number in system = L q – mean number in queue = L-ρ (not L-1) There are usually fewer in the system than L, and fewer in line than L q, because the probability of n in system is skewed.

Mean and median in system Median number in system = ln(.5)/ln(ρ) - 1 Median number waiting = ln(.5)/ln(ρ) – 2 – If ρ = 2/3 L = 2 Median is 0.7 Lq = 1.33 Median queue is 0.

System time and wait time Customers’ average time through system W = L/λ = 1/(μ-λ) Customers’ average wait to be served W q = ρW = ρ/(μ-λ) Most customers spend less than these times.

Expand from basic model More than one server – in parallel (one queue to many servers) – in series (queues in series or stages) Modify independent arrival assumption Limited number in system Limited customer population Modify service assumptions Constant service time Stages of service Priority classes, rather than simple FIFO

Multiple parallel servers

M servers ρ = λ/(Mμ)ρ is how busy each server is Probability of 0 in system:

2 servers ρ = λ/(2μ)ρ is how busy each server is Probability of 0 in system:

M servers Probability of n in the system If n ≤ M(P(0))(Mρ) n /n! If n ≥ M(P(0))M M ρ n /M!

2 servers Probability of n in the system If n = 1(P(0))2ρ If n ≥ 2(P(0))4ρ n /2

M servers L q = L = L q + λ/μ W q = L q /λ W = W q + 1/μ

2 servers L q = L = L q + λ/μ W q = L q /λ W = W q + 1/μ

Examples a 2 nd pharmacist Burger King vs. McDonald’s: – 1 line to 2 servers vs. 2 lines to 2 servers. 2 slow servers vs. 1 server who is twice as fast How many seats in the cafeteria? – E.g. 1 customer per minute, 15 min. to eat, 15 seats? – How they save when you eat faster Comfortable chairs?

Cookbook Pdf version – cell references Named cells version

Cookbook contents 1.One server (like assignment 7A) 2.One server, arrivals from limited group 3.One server, limited queue length (“balking”) 4.One server, constant service time 5.Stages of service, queue only at start 6.Parallel servers, one queue (Post Office) 7.Parallel servers, no queue (hotel) 8.Priority classes for arrivals