Queuing block diagram; M/M/4 and D/M/2

Queuing system

A queuing system is a system in which customers arrive for service, wait in a queue if all servers are busy, receive service and depart. It is described by the arrival pattern, the service pattern, the number of servers, capacity and queue discipline.

Block diagram

 Calling     arrivals   +-------+     +----------+
 population ---------->| Queue |---->| Server(s)|--> Departures
 (source)               +-------+     +----------+
                        waiting line   service
 |<----------------- Queuing system ------------>|

Customers who find the system full are turned away (lost). Customers may also refuse to join a long queue (balk) or leave after waiting for some time (renege).

Uses

• Planning the number of tellers, counters, doctors or machines.
• Computer systems: CPU scheduling, printers, disk requests, network routers and packet delay.
• Telephone exchanges and call centres.
• Traffic signals, airports and toll booths.
• Manufacturing lines and inventory handling.
• Estimating waiting time, queue length and utilization, and checking simulation models.

M/M/4/20/2000/FCFS

Following the pattern A/B/c/K/N/D:

• M: Poisson (random) arrivals, exponential inter-arrival times.
• M: exponential service times.
• 4: four parallel servers.
• 20: the system capacity is 20, so at most 20 customers (4 in service and 16 waiting); further arrivals are turned away.
• 2000: the calling population is finite, 2000 possible customers.
• FCFS: first come first served.

D|M|2|LIFO|18

• D: deterministic (constant) inter-arrival time.
• M: exponential service time.
• 2: two servers.
• LIFO: last in first out; the most recently arrived waiting customer is served next.
• 18: system capacity is 18 customers.
• The calling population is not given, so it is taken as infinite.

(The order of the last fields differs from the standard Kendall order, but each field is identified by its meaning.)