Queuing model, Kendall, M/D/8/15/LIFO

Model of a queuing system

A queuing model represents a system in which customers arrive from a calling population, join a queue if all servers are busy, are served by one or more servers and then depart. It is defined by its arrival process, service time distribution, number of servers, system capacity and queue discipline. The model is analysed mathematically (for simple cases) or simulated to find waiting time, queue length and utilisation.

 Arrivals      +-----------------+     +--------+   Departures
 (calling  --> |  Queue (waiting | --> | Server | -->
 population)   |  line, buffer)  |     | (s)    |
               +-----------------+     +--------+
                  queue discipline      service
                  (FIFO, LIFO...)       time dist.

Kendall's notation

Kendall's notation is a standard short code, A/B/c/K/N/Z, that names the type of a queue.

If the last fields are left out, the defaults are infinite capacity, infinite population and FIFO. Example: M/M/1 is a single-server queue with Poisson arrivals and exponential service.

Meaning of M/D/8/15/LIFO

• M: arrivals are Poisson (exponential inter-arrival times).
• D: service time is deterministic (constant for every customer).
• 8: eight identical servers working in parallel.
• 15: system capacity is 15, meaning at most 15 customers can be in the system (8 in service and 7 waiting). A 16th arrival is turned away.
• LIFO: the last customer to arrive is served first when a server becomes free.
• Calling population is not written, so it is infinite.

Example: a warehouse with eight loading bays, a fixed loading time per truck and room for only 15 trucks in the yard; the newest truck in the yard is handled first.