Kendall notation

The Kendalls notation allows the standardized description of a queuing system . It was developed by David George Kendall and has largely become the standard. The characteristic parameters of the waiting system are classified in a defined sequence of letters and numbers.

${\displaystyle A/S/m/B/K/SD}$ or simplified , if standard values ​​(see below) are assumed for the remaining parameters. ${\displaystyle A/S/m}$

• ${\displaystyle A}$

stands for arrival process and describes the statistical distribution of the inter- arrival times of arrivals.

• ${\displaystyle S}$

stands for service process and describes the statistical distribution of service times and how long a service unit is used.

Abbreviations for the distributions are used for both the arrival process and the service process. B .:

• ${\displaystyle M}$= Exponential distribution (Markovian Distribution)
• ${\displaystyle D}$ = Constant (deterministic distribution),
• ${\displaystyle H}$= Hyperexponential distribution (linear combination of two or more exponential distributions),
• ${\displaystyle E_{k}}$= Erlang distribution ,
• ${\displaystyle PH}$= Phase distribution
• ${\displaystyle G}$or = General (Independent) Distribution ${\displaystyle GI}$

In some cases, these distributions are extended by further parameters, which are then specified as superscript suffixes (e.g. to identify group arrivals)

• ${\displaystyle m}$

stands for the number of (identical) service units ( ) ${\displaystyle s\geq 1}$

• ${\displaystyle B}$

stands for the capacity (seats) of the queue. (Some authors relate this size to the capacity of the entire waiting system). This parameter also serves to distinguish between (pure) waiting systems and loss systems . If no information is given, the following applies: ${\displaystyle B=\infty }$

• ${\displaystyle K}$

stands for the population size, i.e. H. the maximum number of customers that can arrive at the system. If no information is given, the following applies: ${\displaystyle K=\infty }$

• ${\displaystyle SD}$

stands for the handling discipline (service discipline) z. B .:

• FIFO ( First In - First Out ) or FCFS (First-Come First-Served)
• LIFO ( Last In - First Out ) or LCFS (Last-Come First-Serve)
• SJN ( Shortest-Job-Next ) or SPT (Shortest-Processing-Time)
• RANDOM (random) or SIRO (Serve In Random Order)

If this information is missing, the following applies: FIFO

Examples

• ${\displaystyle M/M/1}$, in detail ${\displaystyle M/M/1/\infty /\infty /FIFO}$

A waiting system with a Poisson arrival process (exponential distribution of the waiting time between the arrival of the tasks), exponentially distributed service time and a service unit. The population and queue length are infinite, the handling discipline is FCFS.

• ${\displaystyle D/M/2/10/\infty /LIFO}$

A waiting system with constant arrival times, exponentially distributed service time, two service units, a queue capacity of 10, an infinitely large population and the LIFO dispatch principle.