M/M/1 Queue
===========

See the :mod:`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 :math:`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.