Network Queueing System Design

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

Model: Network Queueing System Design (Network)

Description

This model represents a communication system where arriving messages are routed through a network based on chosen routing percentages. There are \(N\) random messages that arrive following a Poisson process with a rate of \(λ\) that need to go to a particular destination, and there are \(n\) networks available to process these messages. When a message arrives there is a \(p_i%\) chance that it will be processed by network \(i\). The per message processing cost is \(c_1, c_2,..., c_i\) depending on which network the message is routed through. It also takes time for a message to go through a network. This transit time is denoted by \(S_i\) for each network \(i\) and \(S_i\) follows a triangular distribution with lower limit \(a_i\), upper limit \(b_i\), and mode \(c_i\). Each network behaves like a single-server queue with first-in-first-out service discipline.There is a cost for the length of time a message spends in network \(i\) measured by \(c_i\) per unit of time.

Sources of Randomness

  1. Interarrival time of a message.

  2. The network a message is routed to.

  3. The transit time of a message; depends on the network.

Model Factors

  • process_prob: Probability that a message will go through a particular network i.

    • Default: [0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1]

  • cost_process: Message processing cost of network i.

    • Default: [1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10]

  • cost_time: Cost for the length of time a message spends in a network i per unit of time.

    • Default: [0.005, 0.005, 0.005, 0.005, 0.005, 0.005, 0.005, 0.005, 0.005, 0.005]

  • mode_transit_time: Mode time of transit for network i following a triangular distribution.

    • Default: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

  • lower_limits_transit_time: Lower limits for the triangular distribution for the transit time.

    • Default: [0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5]

  • upper_limits_transit_time: Upper limits for the triangular distribution for the transit time.

    • Default: [1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5, 10.5]

  • arrival_rate: Arrival rate of messages following a Poisson process.

    • Default: 1

  • n_messages: Number of messages that arrive and need to be routed.

    • Default: 1000

  • n_networks: Number of networks.

    • Default: 10

Responses

  • total_cost: Total cost spent to route and process all messages.

References

Barton, R. R., & Meckesheimer, M. (2006). Metamodel-Based Simulation Optimization. S.G. Henderson and B.L. Nelson (Eds.), Handbook in OR & MS, Vol. 13.

Optimization Problem: Minimize Total Cost (NETWORK-1)

Decision Variables

  • process_prob

Objectives

The objective is to minimize total costs, the sum of time costs and network costs for all messages.

Constraints

  • \(0 \le p_i \le 1\) for all \(i = 1, 2, ..., n\)

  • \(\sum_{i=1}^n p_i = 1\)

\(p_1, p_2,..., p_n \in [0, 1]\) are the routing probabilities.

Problem Factors

  • initial_solution: Initial solution from which solvers start.

    • Default: [0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1]

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

    • Default: 1000

Fixed Model Factors

N/A

Starting Solution

  • process_prob: [0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1]

Random Solutions

Generate allocations uniformly at random from the set of vectors of length equal to the number of networks whose values are greater than 0 and less than 1.

Optimal Solution

Unknown

Optimal Objective Function Value

Unknown