Simulating Single server queuing models Simulating Single server

Simulating Single server queuing models

Simulating Single server queuing models • Consider the following sequence of activities that each customer undergoes: 1. Customer arrives 2. Customer waits for service if the server is busy. 3. Customer receives service. 4. Customer departs the system.

Analytical Solutions • Analytical solutions for W, L, Wq, Lq exist However, analytical solution exist at infinity which cannot be reached. • Therefore, Simulation is a most.

Flowchart of an arrival event An Arrival Idle Busy Status of Server Customer enters service Customer joins queue More

Flowchart of a Departure event A Departure NO Queue Empty ? Remove customer from Queue and begin service Yes Set system status to idle More

An example of a hand simulation • Consider the following IAT’s and ST’s: • A 1=0. 4, A 2=1. 2, A 3=0. 5, A 4=1. 7, A 5=0. 2, A 6=1. 6, A 7=0. 2, A 8=1. 4, A 9=1. 9, … • S 1=2. 0, S 2=0. 7, S 3=0. 2, S 4=1. 1, S 5=3. 7, S 6=0. 6 • Want: Average delay in queue • Utilization

Initialization Time = 0 System state Server 0 Clock 0 Server status system 0 # in que Times of Arrival A 0. 4 D 999. Eventlist 0 Time 0 0 Of Last Number Total event delayed delay 0 Area Under Q(t) Statistical Counters 0 Area Under B(t)

Arrival Time = 0. 4 System state 0. 4 Clock 0. 4 system 1 Server status 0 # in que Times of Arrival A 1. 6 D 2. 4 Eventlist 0. 4 Time 0 1 Of Last Number Total event delayed delay 0 Area Under Q(t) Statistical Counters 0 Area Under B(t) A 1=0. 4, A 2=1. 2, A 3=0. 5, A 4=1. 7, A 5=0. 2, A 6=1. 6, A 7=0. 2, A 8=1. 4 S 1=2. 0, S 2=0. 7, S 3=0. 2, S 4=1. 1, S 5=3. 7, S 6=0. 6

Arrival Time = 1. 6 System state 1. 6 Clock 0. 4 1 1. 6 system Server status 1 # in que Times of Arrival A 2. 1 D 2. 4 Eventlist 1. 6 Time 0 1 Of Last Number Total event delayed delay 0 Area Under Q(t) Statistical Counters 1. 2 Area Under B(t) A 1=0. 4, A 2=1. 2, A 3=0. 5, A 4=1. 7, A 5=0. 2, A 6=1. 6, A 7=0. 2, A 8=1. 4 S 1=2. 0, S 2=0. 7, S 3=0. 2, S 4=1. 1, S 5=3. 7, S 6=0. 6

Arrival Time = 2. 1 System state 1. 6 2. 1 0. 4 1 1. 6 2. 1 System 2. 1 Clock Server status 2 # in que Times of Arrival A 3. 8 D 2. 4 Eventlist 2. 1 Time 0 1 Of Last Number Total event delayed delay 0. 5 Area Under Q(t) Statistical Counters 1. 7 Area Under B(t) A 1=0. 4, A 2=1. 2, A 3=0. 5, A 4=1. 7, A 5=0. 2, A 6=1. 6, A 7=0. 2, A 8=1. 4 S 1=2. 0, S 2=0. 7, S 3=0. 2, S 4=1. 1, S 5=3. 7, S 6=0. 6

Departure Time = 2. 4 System state 2. 1 2. 4 Clock 1. 6 1 2. 1 System Server status 1 # in que Times of Arrival A 3. 8 D 3. 1 Eventlist 2. 4 Time 2 0. 8 Of Last Number Total event delayed delay 1. 1 Area Under Q(t) Statistical Counters 2. 0 Area Under B(t) A 1=0. 4, A 2=1. 2, A 3=0. 5, A 4=1. 7, A 5=0. 2, A 6=1. 6, A 7=0. 2, A 8=1. 4 S 1=2. 0, S 2=0. 7, S 3=0. 2, S 4=1. 1, S 5=3. 7, S 6=0. 6

Departure Time = 3. 1 System state 3. 1 Clock 2. 1 1 Server status System 0 # in que Times of Arrival A 3. 8 D 3. 3 Eventlist 3. 1 Time 3 1. 8 Of Last Number Total event delayed delay 1. 8 Area Under Q(t) Statistical Counters 2. 7 Area Under B(t) A 1=0. 4, A 2=1. 2, A 3=0. 5, A 4=1. 7, A 5=0. 2, A 6=1. 6, A 7=0. 2, A 8=1. 4 S 1=2. 0, S 2=0. 7, S 3=0. 2, S 4=1. 1, S 5=3. 7, S 6=0. 6

Departure Time = 3. 1 System state 3. 3 Clock 0 Server status System 0 # in que Times of Arrival A 3. 8 D 999. Eventlist 3. 3 Time 3 1. 8 Of Last Number Total event delayed delay 1. 8 Area Under Q(t) Statistical Counters 2. 9 Area Under B(t) A 1=0. 4, A 2=1. 2, A 3=0. 5, A 4=1. 7, A 5=0. 2, A 6=1. 6, A 7=0. 2, A 8=1. 4 S 1=2. 0, S 2=0. 7, S 3=0. 2, S 4=1. 1, S 5=3. 7, S 6=0. 6

Departure Time = 3. 1 System state 3. 8 Clock 3. 8 1 Server status System 0 # in que Times of Arrival A 4. 0 D 4. 9 Eventlist 3. 8 Time 4 1. 8 Of Last Number Total event delayed delay 1. 8 Area Under Q(t) Statistical Counters 2. 9 Area Under B(t) A 1=0. 4, A 2=1. 2, A 3=0. 5, A 4=1. 7, A 5=0. 2, A 6=1. 6, A 7=0. 2, A 8=1. 4 S 1=2. 0, S 2=0. 7, S 3=0. 2, S 4=1. 1, S 5=3. 7, S 6=0. 6

Departure Time = 3. 1 System state 4. 0 3. 8 4. 0 System 1 Server status 1 # in que Times of Arrival 4. 0 Clock A 5. 6 D 4. 9 Eventlist 4. 0 Time 4 1. 8 Of Last Number Total event delayed delay 1. 8 Area Under Q(t) Statistical Counters 3. 1 Area Under B(t) A 1=0. 4, A 2=1. 2, A 3=0. 5, A 4=1. 7, A 5=0. 2, A 6=1. 6, A 7=0. 2, A 8=1. 4 S 1=2. 0, S 2=0. 7, S 3=0. 2, S 4=1. 1, S 5=3. 7, S 6=0. 6

Departure Time = 3. 1 System state 4. 9 Clock 4. 0 1 Server status System 0 # in que Times of Arrival A 5. 6 D 8. 6 Eventlist 4. 9 Time 5 2. 7 Of Last Number Total event delayed delay 2. 7 4. 0 Area Under Q(t) B(t) Statistical Counters A 1=0. 4, A 2=1. 2, A 3=0. 5, A 4=1. 7, A 5=0. 2, A 6=1. 6, A 7=0. 2, A 8=1. 4 S 1=2. 0, S 2=0. 7, S 3=0. 2, S 4=1. 1, S 5=3. 7, S 6=0. 6

Monte Carlo Simulation • Solving deterministic problems using stochastic models. – Example: estimate • It is efficient in solving multi dimensional integrals.

Monte Carlo Simulation • To illustrate, consider a known region R with area A and R 1 subset of R whose area A 1 in unknown. • To estimate the area of R 1 we can through random points in the region R. The ratio of points in the region R 1 over the points in R approximately equals the ratio of A 1/A. R R 1

Monte Carlo Simulation • To estimate the integral I. one can estimate the area under the curve of g. – Suppose that M = max {g(x) } on [a, b] 1. Select random numbers X 1, X 2, …, Xn in [a, b] M R And Y 1, Y 2, … , Yn in [0, M] R 1 a 2. Count how many points (Xi, Yi) in R 1, say C 1 b 3. The estimate of I is then C 1 M(b-a)/n

Advantages of Simulation • Most complex, real-world systems with stochastic elements that cannot be described by mathematical models. Simulation is often the only investigation possible • Simulation allow us to estimate the performance of an existing system under proposed operating conditions. • Alternative proposed system designs can be compared with the existing system • We can maintain much better control over the experiments than with the system itself • Study the system with a long time frame

Disadvantages of Simulation • Simulation produces only estimates of performance under a particular set of parameters • Expensive and time consuming to develop • The Large volume of numbers and the impact of the realistic animation often create high level of confidence than is justified.

Pitfalls of Simulation • Failure to have a well defined set of objectives at the beginning of the study • Inappropriate level of model details • Failure to communicate with manager during the course of simulation • Treating a simulation study as if it is a complicated exercise in computer programming • Failure to have well trained people familiar with operations research and statistical analysis • Using commercial software that may contain errors

Pitfalls of Simulation cont. • Reliance on simulator that make simulation accessible to anyone • Misuse of animation • Failure to account correctly for sources of randomness in the actual system • Using arbitrary probability distributions as input of the simulation • Do output analysis un correctly • Making a single replication and treating the output as true answers • Comparing alternative designs based on one replication of each design • Using wrong measure of performance