Lecture 4 Mathematical and Statistical Models in Simulation

2 Queueing Models  Simulation is often used in the analysis of queueing models.  Typical measures of system performance include server utilization (percentage of time server is busy), length of waiting lines, and delays of customers.  Decision maker is involved in trade-offs between server utilization and customer satisfaction in terms of line lengths and delays.  Simulation is often used in the analysis of queueing models.  Typical measures of system performance include server utilization (percentage of time server is busy), length of waiting lines, and delays of customers.  Decision maker is involved in trade-offs between server utilization and customer satisfaction in terms of line lengths and delays. Calling populationWaiting lineServer

3  In a single-channel queueing system there are only two possible events that can affect the state of the system.  The entry of a unit into the system or the completion of service on a unit  The server has only two possible states:  it is either busy or idle  In a single-channel queueing system there are only two possible events that can affect the state of the system.  The entry of a unit into the system or the completion of service on a unit  The server has only two possible states:  it is either busy or idle Flow Diagram Departure event Being server idle time Remove the waiting unit from the queue Being servicing the unit Another unit waiting ? NO YES

4 Example 2.1  Single-channel queue serves customers on a first-in, first-out (FIFO) basis Customer Number Arrival Time (Clock) Time Service Begins (Clock) Service Time (Duration) Time Service Ends (Clock) Table 2.4. Simulation Table Emphasizing Clock Times

5 Chronological Order  The occurrence of the two types of events Event TypeCustomer NumberClock Time Arrival10 Departure12 Arrival22 Departure23 Arrival36 47 Departure39 Arrival59 Departure411 Departure512 Arrival615 Departure619 Table 2.5. Chronological Ordering of Events

6 Chronological Ordering (cont’)  Number in system at time t Number of customers in the system

7Terminology  mean arrival rate (number of calling units per unit of time)   mean service rate of one server (number of calling units served per unit of time)  1/  mean service time for a calling unit  snumber of parallel service facilities in the system  L q mean length of the queue  Lmean number in the system (those in queue + being served)  W q mean time spent waiting in the queue  Wmean time spent in the system (W q + 1/  )   server utilization factor  mean arrival rate (number of calling units per unit of time)   mean service rate of one server (number of calling units served per unit of time)  1/  mean service time for a calling unit  snumber of parallel service facilities in the system  L q mean length of the queue  Lmean number in the system (those in queue + being served)  W q mean time spent waiting in the queue  Wmean time spent in the system (W q + 1/  )   server utilization factor

8 Statistical Models in Simulation  Discrete Distribution – Poisson ( ) estimate “number of arrivals per unit time” where P(x) = the probability of X successes given a knowledge of = expected number of successes e = mathematical constant approximated by x = number of successes per unit  Discrete Distribution – Poisson ( ) estimate “number of arrivals per unit time” where P(x) = the probability of X successes given a knowledge of = expected number of successes e = mathematical constant approximated by x = number of successes per unit

9 Poisson Distribution  Def: N(t) is a Possion process if  Arrivals occurs individually (at rate )  N(t) has stationary increments: The distribution of the numbers of arrivals between t and t+s depends on the length of the interval s and not on the starting point t.  N(t) has independent increments: The numbers of arrivals during nonoverlapping time intervals (t, t+s) and (t’, t’+s’) are independent random variables.  Def: N(t) is a Possion process if  Arrivals occurs individually (at rate )  N(t) has stationary increments: The distribution of the numbers of arrivals between t and t+s depends on the length of the interval s and not on the starting point t.  N(t) has independent increments: The numbers of arrivals during nonoverlapping time intervals (t, t+s) and (t’, t’+s’) are independent random variables.

10 Uniform Distribution  Continuous Distribution – Uniform distribution A random variable x is uniformly distributed on the interval (a, b):  Continuous Distribution – Uniform distribution A random variable x is uniformly distributed on the interval (a, b):

11 Uniform Distribution (cont’)  The uniform distribution plays a vital role in simulation. Random numbers, uniformly distribution between zero to 1, provide the means to generate random events.

12 Exponential Distribution  Continuous Distribution – Exponential distribution has been used to model interarrival times when arrivals are completely random and to model service times which are highly variable. A random variable x is exponentially distributed with parameter >0:  Continuous Distribution – Exponential distribution has been used to model interarrival times when arrivals are completely random and to model service times which are highly variable. A random variable x is exponentially distributed with parameter >0:

13Memoryless  Memoryless

14 Example of Memoryless  Suppose that the life of an industrial lamp, in thousands of hours, is exponentially distributed with failure rate =1/3 (one failure every 3000 hours, on the average). Find the probability that the industrial lamp will last for another 1000 hours, given that it is operating after 2500 hours.