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).