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Merge pull request #54 from ajkeith/patch-1
Create basic M/M/c queue example
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‎docs/src/examples/mmc.md

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# Multi-server Queue
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## Description
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An [M/M/c queue](https://en.wikipedia.org/wiki/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 _λ_ such that the interarrival time distribution has mean _1/λ_. Similarly, the service rate is defined as _μ_ such that the service time distribution has mean _1/μ_ (for each server). The overall traffic intensity of the queue is _ρ = λ / (c * μ)_. If the traffic intensity exceeds one, the queue is unstable and the queue length will grow indefinitely.
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## Code
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```julia
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#set simulation parameters
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srand(8710) # set random number seed for reproducibility
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num_customers = 10 # total number of customers generated
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# set queue parameters
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num_servers = 2 # number of servers
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mu = 1.0 / 2 # service rate
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lam = 0.9 # arrival rate
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arrival_dist = Exponential(1 / lam) # interarrival time distriubtion
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service_dist = Exponential(1 / mu) # service time distribution
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# define customer behavior
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@resumable function customer(env::Environment, server::Resource, id::Integer, t_a::Float64, d_s::Distribution)
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@yield timeout(env, t_a) # customer arrives
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println("Customer $id arrived: ", now(env))
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@yield request(server) # customer starts service
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println("Customer $id entered service: ", now(env))
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@yield timeout(env, rand(d_s)) # server is busy
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@yield release(server) # customer exits service
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println("Customer $id exited service: ", now(env))
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end
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# setup and run simulation
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sim = Simulation() # initialize simulation environment
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server = Resource(sim, num_servers) # initialize servers
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arrival_time = 0.0
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for i = 1:num_customers # initialize customers
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arrival_time += rand(arrival_dist)
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@process customer(sim, server, i, arrival_time, service_dist)
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end
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run(sim) # run simulation
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## output
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#
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# Customer 1 arrived: 0.1229193244813443
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# Customer 1 entered service: 0.1229193244813443
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# Customer 2 arrived: 0.22607641035584877
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# Customer 2 entered service: 0.22607641035584877
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# Customer 3 arrived: 0.4570009029409502
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# Customer 2 exited service: 1.7657345101378559
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# Customer 3 entered service: 1.7657345101378559
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# Customer 1 exited service: 2.154824561031012
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# Customer 3 exited service: 2.2765287086137764
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# Customer 4 arrived: 2.3661687470062995
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# Customer 4 entered service: 2.3661687470062995
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# Customer 5 arrived: 2.6110816119637885
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# Customer 5 entered service: 2.6110816119637885
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# Customer 5 exited service: 2.8017888690417583
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# Customer 6 arrived: 3.019540357955037
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# Customer 6 entered service: 3.019540357955037
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# Customer 6 exited service: 3.351151832298383
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# Customer 7 arrived: 3.5254699872847612
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# Customer 7 entered service: 3.5254699872847612
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# Customer 7 exited service: 4.261422043181396
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# Customer 4 exited service: 4.602071952938201
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# Customer 8 arrived: 7.27536704811686
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# Customer 8 entered service: 7.27536704811686
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# Customer 9 arrived: 7.491176033637809
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# Customer 9 entered service: 7.491176033637809
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# Customer 10 arrived: 8.39098457094977
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# Customer 8 exited service: 8.683396356977969
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# Customer 10 entered service: 8.683396356977969
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# Customer 9 exited service: 8.7501656586875
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# Customer 10 exited service: 9.049670951561666
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```

‎examples/queue_mmc.ipynb

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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Multi-server Queue\n",
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"\n",
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"## Description \n",
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"An [M/M/c queue](https://en.wikipedia.org/wiki/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. "
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Install Packages"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"Pkg.add(\"Distributions\")\n",
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"Pkg.add(\"SimJulia\")"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Load Packages"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mPrecompiling module Distributions.\n",
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"\u001b[39m\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mPrecompiling module SimJulia.\n",
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"\u001b[39m"
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]
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}
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],
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"source": [
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"using Distributions\n",
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"using SimJulia, ResumableFunctions"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Define Constants"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"srand(8710) # set random number seed for reproducibility\n",
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"num_customers = 10 # total number of customers generated\n",
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"num_servers = 2 # number of servers\n",
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"mu = 1.0 / 2 # service rate\n",
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"lam = 0.9 # arrival rate\n",
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"arrival_dist = Exponential(1 / lam) # interarrival time distriubtion\n",
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"service_dist = Exponential(1 / mu); # service time distribution"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Define Customer Behavior"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"customer (generic function with 1 method)"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"@resumable function customer(env::Environment, server::Resource, id::Integer, time_arr::Float64, dist_serve::Distribution)\n",
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" @yield timeout(env, time_arr) # customer arrives\n",
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" println(\"Customer $id arrived: \", now(env))\n",
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" @yield request(server) # customer starts service\n",
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" println(\"Customer $id entered service: \", now(env))\n",
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" @yield timeout(env, rand(dist_serve)) # server is busy\n",
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" @yield release(server) # customer exits service\n",
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" println(\"Customer $id exited service: \", now(env))\n",
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"end"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Setup and Run Simulation"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Customer 1 arrived: 0.1229193244813443\n",
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"Customer 1 entered service: 0.1229193244813443\n",
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"Customer 2 arrived: 0.22607641035584877\n",
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"Customer 2 entered service: 0.22607641035584877\n",
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"Customer 3 arrived: 0.4570009029409502\n",
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"Customer 2 exited service: 1.7657345101378559\n",
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"Customer 3 entered service: 1.7657345101378559\n",
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"Customer 1 exited service: 2.154824561031012\n",
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"Customer 3 exited service: 2.2765287086137764\n",
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"Customer 4 arrived: 2.3661687470062995\n",
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"Customer 4 entered service: 2.3661687470062995\n",
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"Customer 5 arrived: 2.6110816119637885\n",
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"Customer 5 entered service: 2.6110816119637885\n",
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"Customer 5 exited service: 2.8017888690417583\n",
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"Customer 6 arrived: 3.019540357955037\n",
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"Customer 6 entered service: 3.019540357955037\n",
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"Customer 6 exited service: 3.351151832298383\n",
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"Customer 7 arrived: 3.5254699872847612\n",
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"Customer 7 entered service: 3.5254699872847612\n",
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"Customer 7 exited service: 4.261422043181396\n",
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"Customer 4 exited service: 4.602071952938201\n",
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"Customer 8 arrived: 7.27536704811686\n",
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"Customer 8 entered service: 7.27536704811686\n",
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"Customer 9 arrived: 7.491176033637809\n",
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"Customer 9 entered service: 7.491176033637809\n",
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"Customer 10 arrived: 8.39098457094977\n",
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"Customer 8 exited service: 8.683396356977969\n",
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"Customer 10 entered service: 8.683396356977969\n",
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"Customer 9 exited service: 8.7501656586875\n",
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"Customer 10 exited service: 9.049670951561666\n"
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]
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}
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],
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"source": [
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"sim = Simulation() # initialize simulation environment\n",
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"server = Resource(sim, num_servers) # initialize servers\n",
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"arrival_time = 0.0\n",
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"for i = 1:num_customers # initialize customers\n",
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" arrival_time += rand(arrival_dist)\n",
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" @process customer(sim, server, i, arrival_time, service_dist)\n",
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"end\n",
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"run(sim) # run simulation"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Julia 0.6.0",
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"language": "julia",
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"name": "julia-0.6"
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},
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"language_info": {
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"file_extension": ".jl",
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"mimetype": "application/julia",
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"name": "julia",
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"version": "0.6.0"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}

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