# Discrete Event Simulation: Queues and Servers

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## WHAT IS IT?

This is a simple queueing system model, with a single, unlimited queue and 1-10 homogeneous servers. Arrivals follow a Poisson process, and service times are exponentially distributed.

## HOW IT WORKS

This is a discrete-event simulation, which is a type of simulation that advances the clock in discrete, often irregularly sized steps, rather than by very small, regular time slices (which are generally used to produce quasi-continuous simulation). At each step, the clock is advanced to the next scheduled event in an event queue, and that event is processed. In this model, the different events are: customer arrival and entry into the queue (followed, if possible, by start of service); service completion, with the customer leaving the system (followed, if possible, by start of service for a new customer); statistics reset; and simulation end. Since these are the only events that can result in a change of the state of the simulation, there is no point in advancing the clock in smaller time steps than the intervals between the events.

## HOW TO USE IT

Use the **number-of-servers** slider to set the number of servers; then press the **Setup** button to create the servers and reset the simulation clock.

The **mean-arrival-rate** and **mean-service-time** sliders control the arrival and service processes, respectively. These values can be changed before starting the simulation, or at anytime during the simulation run; any changes are reflected immediately in a running model.

The **max-run-time** and **stats-reset-time** sliders control the length of the simulation and the time at which all the aggregate statistics are reset, respectively. The latter allows for reducing the effects of system startup on the aggregate statistics.

The simulation can be run one step at a time with the **Next** button, or by repeatedly processing events with the **Go!** button.

The aggregate statistics can be reset at any time – without emptying the queue or placing servers in the idle state – with the **Reset Stats** button.

## THINGS TO NOTICE

After the simulation has started, the next scheduled arrival time is always shown in the **Next Arrival Time** monitor. When any of the servers are busy, the scheduled time of service completion is shown in the label below the server.

In queueing theory notation, the type of system being simulated in this model is referred to as *M/M/n* – i.e. Poisson arrivals, exponential service times, infinite queue capacity and source population, FIFO queue discipline. When there is a single server, or when all the servers have the same mean service time, the steady state characteristics (if the system is capable of reaching a steady state) can be determined analytically. In this model, these theoretical values are shown in the bottom row of monitors. If the theoretical server utilization – determined by multiplying the arrival rate by the service time, dividing by the number of servers, and taking the lesser the result of the calculation and 1 – is less than 1, then the queueing equations have a defined solution; otherwise, the expected queue length and expected time in the queue are unbounded. In this model, these unbounded values are denoted by "N/A" in the associated monitors.

This model displays servers in a row along the bottom of the NetLogo world; customers are shown in a queue which "snakes" from near the bottom of the NetLogo world to the top. However, these display features are purely for visualization purposes; the positions of the servers and customers, and the colors of the customers, have no functional purpose or impact. The colors of the servers, on the other hand, does have a meaning: an idle server is shown in green, while a busy server is red.

## THINGS TO TRY

Run the simulation several times, to get a sense of the effects of the different parameters on the average queue length and average time in the queue. How do these observed statistics compare with the theoretical values? Do the input parameters seem to affect not only the average queue length, but also the variability of the queue length?

## EXTENDING THE MODEL

This model could easily be extended to support non-identical mean service times for different servers (possibly through an **Add Server** button that creates servers one at a time, each with a specified mean service time value); additional service time distributions besides exponential; a capacitated queue; and alternative queue disciplines (random priority and LIFO would be the easiest to add). However, when simulating a system with these complicating factors, the computations for expected queue length and expected time in the queue can become difficult, or even practically impossible. Note, however, that there is a general relationship – known as Little's formula – between expected queue length and expected time in the queue (or, more generally, between expected number of customers/transactions in the entire system, and the expected time a customer/transaction spends in the system), which holds for even very complicated queueing systems.

## NETLOGO FEATURES

This model uses the ** tick-advance** primitive to advance the NetLogo ticks value by non-integral amounts. This allows the NetLogo clock to be used as a discrete-event simulation clock. However, the standard ticks display (normally seen just below the speed slider) is unable to display non-integral values, so this model uses a separate ticks monitor.

## CREDITS AND REFERENCES

Copyright 2017 Nick Bennett

Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at

http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.

## Comments and Questions

;;------------------------------------------------------------------------------ ;; Copyright 2017 Nick Bennett ;; ;; Licensed under the Apache License, Version 2.0 (the "License"); ;; you may not use this file except in compliance with the License. ;; You may obtain a copy of the License at ;; ;; http://www.apache.org/licenses/LICENSE-2.0 ;; ;; Unless required by applicable law or agreed to in writing, software ;; distributed under the License is distributed on an "AS IS" BASIS, ;; WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. ;; See the License for the specific language governing permissions and ;; limitations under the License. ;;------------------------------------------------------------------------------ ;;------------------------------------------------------------------------------ ;; Version History ;; 2010-02-21 - v1.0: M/M/n queueing simulation; up to 10 servers; standard ;; aggregate statistics & corresponding expected values. ;; 2017-07-26 - v1.5: Updated for NetLogo 6. Licensed under Apache License ;; 2.0. ;; 2017-07-29 - v1.6: Added plot & more NetLogo 6-related updates, including ;; use of anonymous procedures and range reporter. ;; 2017-08-03 - v1.6.1: Fix info details. ;;------------------------------------------------------------------------------ ;; The model has two types of agents: customers, who enter and wait in a queue ;; until a server is available; and servers, who serve each customer in ;; first-come, first-served order. breed [customers customer] breed [servers server] ;; Each customer records the time entering the system, and the time entering ;; service, so that average time-in-queue/time-in-system statistics can be ;; computed. customers-own [ time-entered-queue time-entered-service ] ;; Each server records the customer agent being served, and the scheduled ;; completion of that service. Since servers are homogenous, individual ;; utilization statistics aren't kept. servers-own [ customer-being-served next-completion-time ] globals [ ; Waiting line queue ; Arrival process arrival-count next-arrival-time ; Statistics for average load/usage of queue and servers stats-start-time total-customer-queue-time total-customer-service-time ; Statistics for average time-in-queue and time-in-service total-time-in-queue total-time-in-system total-queue-throughput total-system-throughput ; Physical layout parameters server-ycor queue-server-offset customer-xinterval customer-yinterval customers-per-row ; Saved slider values to allow detection of changes during a simulation run save-mean-arrival-rate save-mean-service-time ; Theoretical measures, computed analytically using classic queueing theory expected-utilization expected-queue-length expected-queue-time ; Anonymous procedures end -run-task arrive-task complete-service-task reset-stats-task ] ;; Sets up the model when loaded. to startup setup end ;; Initializes global variables and server agents. to setup clear-all setup-globals setup-servers setup-tasks compute-theoretical-measures reset-ticks reset-stats schedule-arrival end ;; Resets statistics, initializes queue list, and sets agent shapes and other ;; display properties. to setup-globals set queue [] set next-arrival-time 0 set arrival-count 0 set-default-shape servers "server" set-default-shape customers "person" set server-ycor (min-pycor + 1) set queue-server-offset 1.5 set customer-xinterval 0.5 set customer-yinterval 1 set customers-per-row (1 + (world-width - 1) / customer-xinterval) set save-mean-arrival-rate mean-arrival-rate set save-mean-service-time mean-service-time end ;; Creates server agents and arranges them horizontally, evenly spaced along ;; the bottom of the NetLogo world. This layout is purely cosmetic, and has no ;; functional purpose or impact. to setup-servers let horizontal-interval (world-width / number-of-servers) create-servers number-of-servers [ set color green setxy (min-pxcor - 0.5 + horizontal-interval * (0.5 + who)) server-ycor set size 2.75 set label "" set customer-being-served nobody set next-completion-time 0 ] end ;; Sets up anonymous procedures for event queue entries. to setup-tasks set end-run-task [[?ignore] -> end-run] set arrive-task [[?ignore] -> arrive] set complete-service-task [[?server] -> complete-service ?server] set reset-stats-task [[?ignore] -> reset-stats] end ;; Updates statistics (which also advances the clock) and invokes the ;; relevant anonymous procedure for the next event scheduled. to go ifelse (ticks < max-run-time) [ let next-event [] let event-queue (list (list max-run-time end-run-task nobody)) let next-server-to-complete next-server-complete set event-queue ( fput (list next-arrival-time arrive-task nobody) event-queue) if (is-turtle? next-server-to-complete) [ set event-queue (fput (list ([next-completion-time] of next-server-to-complete) complete-service-task next-server-to-complete) event-queue) ] if (stats-reset-time > ticks) [ set event-queue ( fput (list stats-reset-time reset-stats-task nobody) event-queue) ] set event-queue (sort-by [[?1 ?2] -> first ?1 < first ?2] event-queue) set next-event (first event-queue) update-usage-stats (first next-event) set next-event (but-first next-event) (run (first next-event) (last next-event)) ] [ stop ] end ;; Ends the execution of the simulation. In fact, this procedure does nothing, ;; but is still necessary. When the associated event is the first in the event ;; queue, the clock will be updated to the simulation end time prior to this ;; procedure being invoked; this causes the go procedure to stop on the next ;; iteration. to end-run ; Do nothing end ;; Creates a new customer agent, adds it to the queue, and attempts to start ;; service. to arrive let color-index (arrival-count mod 70) let main-color (floor (color-index / 5)) let shade-offset (color-index mod 5) create-customers 1 [ set color (3 + shade-offset + main-color * 10) set time-entered-queue ticks move-forward length queue set queue (lput self queue) set time-entered-queue ticks ] set arrival-count (arrival-count + 1) schedule-arrival begin-service end ;; Samples from the exponential distribution to schedule the time of the next ;; customer arrival in the system. to schedule-arrival set next-arrival-time (ticks + random-exponential (1 / mean-arrival-rate)) end ;; If there are customers in the queue, and at least one server is idle, starts ;; service on the first customer in the queue, using a randomly selected ;; idle server, and generating a complete-service event with a time sampled ;; from the exponential distribution. Updates the queue display, moving each ;; customer forward. to begin-service let available-servers (servers with [not is-agent? customer-being-served]) if (not empty? queue and any? available-servers) [ let next-customer (first queue) let next-server one-of available-servers set queue (but-first queue) ask next-customer [ set time-entered-service ticks set total-time-in-queue (total-time-in-queue + time-entered-service - time-entered-queue) set total-queue-throughput (total-queue-throughput + 1) move-to next-server ] ask next-server [ set customer-being-served next-customer set next-completion-time (ticks + random-exponential mean-service-time) set label precision next-completion-time 3 set color red ] (foreach queue (range (length queue)) [ [?1 ?2] -> ask ?1 [ move-forward ?2 ] ]) ] end ;; Updates time-in-system statistics, removes current customer agent, returns ;; the server to the idle state, and attempts to start service on another ;; customer. to complete-service [?server] ask ?server [ set total-time-in-system (total-time-in-system + ticks - [time-entered-queue] of customer-being-served) set total-system-throughput (total-system-throughput + 1) ask customer-being-served [ die ] set customer-being-served nobody set next-completion-time 0 set color green set label "" ] begin-service end ;; Reports the busy server with the earliest scheduled completion. to-report next-server-complete report (min-one-of (servers with [is-agent? customer-being-served]) [next-completion-time]) end ;; Sets all aggregate statistics back to 0 - except for the simulation start ;; time (used for computing average queue length and average server ;; utilization), which is set to the current time (which is generally not 0, ;; for a reset-stats event). to reset-stats set total-customer-queue-time 0 set total-customer-service-time 0 set total-time-in-queue 0 set total-time-in-system 0 set total-queue-throughput 0 set total-system-throughput 0 set stats-start-time ticks end ;; Updates the usage/utilization statistics and advances the clock to the ;; specified event time. to update-usage-stats [event-time] let delta-time (event-time - ticks) let busy-servers (servers with [is-agent? customer-being-served]) let in-queue (length queue) let in-process (count busy-servers) let in-system (in-queue + in-process) set total-customer-queue-time (total-customer-queue-time + delta-time * in-queue) set total-customer-service-time (total-customer-service-time + delta-time * in-process) tick-advance (event-time - ticks) update-plots end ;; Moves to the specified queue position, based on the global spacing ;; parameters. This queue display is purely cosmetic, and has no functional ;; purpose or impact. to move-forward [queue-position] let new-xcor (max-pxcor - customer-xinterval * (queue-position mod customers-per-row)) let new-ycor (server-ycor + queue-server-offset + customer-yinterval * floor (queue-position / customers-per-row)) ifelse (new-ycor > max-pycor) [ hide-turtle ] [ setxy new-xcor new-ycor if (hidden?) [ show-turtle ] ] end ;; Checks to see if the values of the mean-arrival-rate and mean-service-time ;; sliders have changed since the last time that the theoretical system measures ;; were calculated; if so, the theoretical measures are recalculated. ;; (Note: This reporter is invoked by the expected queue length monitor. Since ;; monitors are updated periodically, even when a forever button isn't pressed, ;; this allows the theoretical statistics to be updated when the sliders ;; change, even if the model isn't running. to-report sliders-changed? let changed? false if ((save-mean-arrival-rate != mean-arrival-rate) or (save-mean-service-time != mean-service-time)) [ set changed? true set save-mean-arrival-rate mean-arrival-rate set save-mean-service-time mean-service-time compute-theoretical-measures ] report changed? end ;; Computes the expected utilization, queue length, and time in queue for M/M/n ;; queueing system. to compute-theoretical-measures let balance-factor (mean-arrival-rate * mean-service-time) let n (count servers) ifelse ((balance-factor / n) < 1) [ let k 0 let k-sum 1 let power-product 1 let factorial-product 1 let busy-probability 0 foreach (range 1 n) [ [?] -> set power-product (power-product * balance-factor) set factorial-product (factorial-product * ?) set k-sum (k-sum + power-product / factorial-product) ] set power-product (power-product * balance-factor) set factorial-product (factorial-product * n) set k (k-sum / (k-sum + power-product / factorial-product)) set busy-probability ((1 - k) / (1 - balance-factor * k / n)) set expected-utilization (balance-factor / n) set expected-queue-length (busy-probability * expected-utilization / (1 - expected-utilization)) set expected-queue-time (busy-probability * mean-service-time / (n * (1 - expected-utilization))) ] [ set expected-utilization 1 set expected-queue-length "N/A" set expected-queue-time "N/A" ] end

There are 3 versions of this model.

## Attached files

File | Type | Description | Last updated | |
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Discrete Event Simulation: Queues and Servers.png | preview | Preview for 'Discrete Event Simulation: Queues and Servers' | about 7 years ago, by Nicholas Bennett | Download |