Simulation Statistics Numerous standard statistics of interest Some results calculated from parameters Used to verify the simulation Most calculated by program

Some Statistics Average Wait time for a customer = total time customers wait in queue total number of customers Average wait time of those who wait = total time of customers who wait in queue number of customers who wait

More Statistics Proportion of server busy time = number of time units server busy total time units of simulation Average service Time = total service time number of customers serviced

More Statistics Average time customer spends in system = total time customers spend in system total number of customers Probability a customer has to wait in queue = number of customers who wait

Traffic Intensity A measure of the ability of the server to keep up with the number of the arrivals TI= (service mean)/(inter-arrival mean) If TI > 1 then system is unstable & queue grows without bound

Server Utilization % of time the server is busy serving customers If there is 1 server SU = TI = (service mean)/(inter-arrival mean) If there are N servers SU = 1/N * (service mean)/(inter-arrival mean)

Statistical Models Probability: a quantitive measure of the chance or likelihood of an event occurring. Random: unable to be predicted exactly In an experiment where events randomly occur but in which we have assigned to each possible outcome a probability, we have determined a probability or stochastic model

Terms Event Space Event Complement of an Event Intersection Union Mutually Exclusive

Examples Event Space: The set of all possible events that can occur Event (E): Any single occurrence ex: E = {4,5} Complement of E: Set of all events except E Ex: Complement of E = {1,2,3,6}

Examples Union: Combination of any 2 event sets A= {1,2,3} B = {3,4} A U B = {1,2,3,4}

Examples Intersection: Overlap of common occurrence of 2 event sets A= {1,2,3} B = {3,4} A Π B = {3} Mutually Exclusive: 2 event sets that have no events in common A= {1,2} B = {3,4} A Π B = { }

Random variable Practical Definition a quantity whose value is determined by the outcome of a random experiment

Random Variable Examples X = the number of 4's that occur in 10 rolls Y = the number of customers that arrive in 1 hour Z = the number of services that are completed in 5 minutes

Discrete vs. Continuous RV EXAMPLE Discrete: X = number of customers that arrive in 1 hour Continuous: Y = gallons that flow into the pool in 1 hour ????: Z = the average age of the customers that arrive in an hour

Discrete: Probability Function Let X be a discrete R.V. with possible values x1, x2,…xn. Let P be the probability function P(xi) = (X = xi) such that (a) P(xi) >= 0 for i = 1,2,…n (b) Σ P(xi) = 1

Probability Function Example Consider the rolling of a fair die 1/6 for x = 1 P(x) = 1/6 for x = 2 1/6 for x = 3 1/6 for x = 4 1/6 for x = 5 1/6 for x = 6 0 for all other x

Cumulative Distribution Function CDF of a random variable X is F such that F(x) = P (X <= x) F(X) is continuous Discrete: sum of probabilities Continuous: area under the curve

Cumulative Distribution Function - Example Consider the rolling of a fair die 0 for x < 1 1/6 for x < 2 F(X) = 2/6 for x <3 3/6 for x < 4 4/6 for x < 5 5/6 for x < 6 1 for x >= 6

Cumulative Function 1 1/2 1/6 2 3 4 5 6

Discrete vs. Continuous R.V. Cumulative Distribution Function (CDF) The CDF of a discrete R.V. X is F such that F(x)= P (X<= x) Continuous: The CDF of a continuous RV has the properties: F(x) is continuous, at least piecewise F(x) exists except in at most a finite number of points

Discrete vs. Continuous Random Variables Random variable: a function whose domain is the event space & whose range is some subset of real numbers If a random variable assumes a discrete (finite or countably infinite) number of values, it is called a discrete random variable. Otherwise, it is called a continuous random variable.