Model: M/M/1 Queue

Description:

This is a model simulates an M/M/1 queue with an Exponential interarrival time distribution and an Exponential service time distribution.

Sources of Randomness:

1. Exponential interarrival times.

2. Exponential service times.

Model Factors:

• lambda: Rate parameter of interarrival time distribution.

  • Default: 1.5

• mu: Rate parameter of service time distribution.

  • Default 3.0

• warmup: Represents the number of people as warmup before collecting statistics

  • Default: 50

• people: Represents the number of people from which to calculate the average sojourn time.

  • Default: 200

Respones:

• avg_sojourn_time: The average of sojourn time of customers (time customers spend in the system).

• avg_waiting_time: The average of waiting time of customers.

• frac_cust_wait: The fraction of customers who wait.

References:

This example is adapted from Cheng, R and Kleijnen,J.(1999). Improved Design of Queueing Simulation Experience with Highly Heteroscedastic Responses. Operations Research, v. 47, n. 5, pp. 762-777 (https://pubsonline.informs.org/doi/abs/10.1287/opre.47.5.762)

Optimization Problem: Minimize average sojourn time plus penalty (MM1-1)

Decision Variables:

• mu (service rate parameter)

Objectives:

Minimize the expected average sojourn time plus a penalty for increasing the rate \(c\mu^2\).

Constraints:

No deterministic or stochastic constraints. Box constraints for non-negativity of mu.

Problem Factors:

• budget: Max # of replications for a solver to take.

  • Default: 1000

• cost: Cost for increasing service rate.

  • Default: 0.1

Fixed Model Factors:

None

Starting Solution:

• mu: 3.0

Random Solutions:

Generate mu from an exponential distribution with mean 3.

Optimal Solution:

None.

Optimal Objective Function Value:

None.