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:¶
Interarrival time of a message.
The network a message is routed to.
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