# M/M/1 Solver & Simulator

This document shows two fundamentally different approaches to problem solving: Computation using a Mathematical model and Simulation. Read a brief introduction to Queuing Theory and if you want, there is also a full derivation of the M/M/1 model. Enter your input data and then compute and simulate. It may be interesting to compare the basic results (you are also given additional results from the simulaton). For some systems (like l = 1, m = 2) simulated and computed results are very similar - the differences are caused by random fluctuations and also by a limited length of the simulation experiment. Bur try this: l = 1, m = 1.001. Theoretically the average queue length should be 999. In one particular experiment (length 10000, 9765 arrivals) the maximum queue length was 107, the average about 39. Where is the problem ? Limited precision, but of what ? Random generator or the Log function when computing the exponentially distributed figures ? Too short experiment for this limit case ? This shows that one is never enough careful when interpreting both theoretical and simulation results.

## Input data

Input Parameter The value Explanation
Arrival Rate (l)
Service Rate (m)
Experiment duration
Maximum queue length

## Basic Results

Result Computed value Simulated value Explanation
Customers in system (Ls)
Customers in queue (Lq)
Time in system (Ws)
Time in queue (Wq)
Idle probability (p0)
Server utilization (r)

## Other Simulation Results

Result Value Explanation
Number of arrivals
Minimum arrival interval
Maximum arrival interval
Number of services
Minimum service duration
Maximum service duration
Maximum waiting time
Maximum time in system
Maximum queue length