queue_mmc


Multi-server Queue

Description

An M/M/c queue is a basic queue with $c$ identical servers, exponentially distributed interarrival times, and exponentially distributed service times for each server. The arrival rate is defined as $\lambda$ such that the interarrival time distribution has mean $1/\lambda$. Similarly, the service rate is defined as $\mu$ such that the service time distribution has mean $1/\mu$ (for each server). The overall traffic intensity of the queue is $\rho = \lambda / (c \mu)$. If the traffic intensity exceeds one, the queue is unstable and the queue length will grow indefinitely.

Install Packages


In [ ]:
Pkg.add("Distributions")
Pkg.add("SimJulia")

Load Packages


In [2]:
using Distributions
using SimJulia, ResumableFunctions


INFO: Precompiling module Distributions.
INFO: Precompiling module SimJulia.

Define Constants


In [3]:
srand(8710) # set random number seed for reproducibility
num_customers = 10 # total number of customers generated
num_servers = 2 # number of servers
mu = 1.0 / 2 # service rate
lam = 0.9 # arrival rate
arrival_dist = Exponential(1 / lam) # interarrival time distriubtion
service_dist = Exponential(1 / mu); # service time distribution

Define Customer Behavior


In [4]:
@resumable function customer(env::Environment, server::Resource, id::Integer, time_arr::Float64, dist_serve::Distribution)
    @yield timeout(env, time_arr) # customer arrives
    println("Customer $id arrived: ", now(env))
    @yield request(server) # customer starts service
    println("Customer $id entered service: ", now(env))
    @yield timeout(env, rand(dist_serve)) # server is busy
    @yield release(server) # customer exits service
    println("Customer $id exited service: ", now(env))
end


Out[4]:
customer (generic function with 1 method)

Setup and Run Simulation


In [5]:
sim = Simulation() # initialize simulation environment
server = Resource(sim, num_servers) # initialize servers
arrival_time = 0.0
for i = 1:num_customers # initialize customers
    arrival_time += rand(arrival_dist)
    @process customer(sim, server, i, arrival_time, service_dist)
end
run(sim) # run simulation


Customer 1 arrived: 0.1229193244813443
Customer 1 entered service: 0.1229193244813443
Customer 2 arrived: 0.22607641035584877
Customer 2 entered service: 0.22607641035584877
Customer 3 arrived: 0.4570009029409502
Customer 2 exited service: 1.7657345101378559
Customer 3 entered service: 1.7657345101378559
Customer 1 exited service: 2.154824561031012
Customer 3 exited service: 2.2765287086137764
Customer 4 arrived: 2.3661687470062995
Customer 4 entered service: 2.3661687470062995
Customer 5 arrived: 2.6110816119637885
Customer 5 entered service: 2.6110816119637885
Customer 5 exited service: 2.8017888690417583
Customer 6 arrived: 3.019540357955037
Customer 6 entered service: 3.019540357955037
Customer 6 exited service: 3.351151832298383
Customer 7 arrived: 3.5254699872847612
Customer 7 entered service: 3.5254699872847612
Customer 7 exited service: 4.261422043181396
Customer 4 exited service: 4.602071952938201
Customer 8 arrived: 7.27536704811686
Customer 8 entered service: 7.27536704811686
Customer 9 arrived: 7.491176033637809
Customer 9 entered service: 7.491176033637809
Customer 10 arrived: 8.39098457094977
Customer 8 exited service: 8.683396356977969
Customer 10 entered service: 8.683396356977969
Customer 9 exited service: 8.7501656586875
Customer 10 exited service: 9.049670951561666