I will try to explain some basics of Queueing Theory and how to implement M/M/c in R.

Queueing theory – is the mathematical study of waiting lines, or queues.

Kendall’s notation to describe a queueing system: A / B / m / K / n / D

• A - distribution function of the interarrival times;
• B - distribution function of the service times;
• m - number of servers;
• K - capacity of the system, the maximum number of customers in the system including the one being serviced;
• n - population size, number of sources of customers;
• D - service discipline (FIFO, LIFO, RS – Random Service, Priority, etc.).

We look at M/M/c with FIFO service discipline with:

• Arrival rate – Poisson distribution;
• Service rate – Exponential distribution;

Time rate is important in QT. Time units could be : second, minute, hour, day, etc.

Parameters:

• m – number of servers;
• lambda – arrival rate / time unit;
• mu – service rate / time unit;

Queueing Theory can asnwer following questions:

• Efficiency of each server (p);
• Probability of zero or n customer in the system (p_0, p_n);
• The mean number of customers waiting in queue (L_q);
• Mean time customers spend in queue (W_q);
• Mean time customers spend in system (W);
• Mean number of customers in system (L).