Queue-Simulation

Simulation and stochastic modelling of different types of queues with some analysis.

1 type of customer problem

• M/M/1 queue: a single line served by a single server on a first-come, first-served basis
• n-server queue: a single line served by n servers
• routing methods:
  • index-based: first customer allocated to server 1, next customer to server 2, etc.
  • random: route to a random server
  • longest idle: allocate next customer to longest idle server
• abandonment/reneging: sometimes customers run out of patience and decide to leave the queue without being served
• notation:
  • $\mu$ : mean service rate (e.g. 2/hour)
  • $\lambda$ : mean arrival rate of customers (e.g. 5/hour)
  • $\gamma$ : mean abandonment rate (e.g. 2/hour means each customer has an average patience time of 0.5 hours)
• analysis:
  • important features: utilization, throughput, average queue length...
  • obtain analytical solutions for the steady state probabilities using Chapman-Kolmogorov equation
  • calculate average queue length from steady state probabilites
  • comparison of average queue length from simulation with analytical solutions
  • observe how average queue length changes with the variance of average service rates (e.g. $\mu_i \sim U[1,3]$ or $U[1.5, 2.5]$ or $U[1.9, 2.1]$)

2 types of customer problems

• type I and type II customers arriving
• each server is capable of catering for both types of problems
• servers have different mean service rates for type I and type II problems
• routing to a server can be 3 types: index-based, random, longest-idle
• choosing a type of customer to serve next can be
  • first come, first served
  • random
  • choose a customer from the longest queue
• Question: Does the dependence of type I and type II mean service rates affect average queue length?

Files in the repository:

• MM1 plots: M/M/1 queue
  • step plot of number of customers in the system
  • queue length pmf (i.e. steady state probabilities graph)
  • average queue length vs $\lambda$ / $\mu$
• 3 routing methods with and without abandonment
• average queue length: method to find average queue length for simulation
• analytical solutions: finding average queue length from steady-state probabilities (matrix inversion from Chapman-Kolmogorov)
• 2-type queue: type I and type II customers arriving
  • 1st come, 1st served
  • random
  • choose next customer from longer queue
• correlation-L graphs: find correlation of $\mu{i1}$ and $\mu{i2}$, then graph it against average queue length obtained from simulation
  • two different designs to create mean service rates of varying dependence
  • calculation of average queue lengths in different scenarios