M/M/1 Queue

See the simopt.models.mm1queue module for API details.

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

Responses

  • 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.