Service discipline

A service discipline specifies which customers receive the server's capacity and where newly arriving customers join the queue. First come first served and processor sharing give different sample paths even when their equilibrium populations agree.

In an ordered queue, let ${\gamma }_i(n)$ be the fraction of service capacity assigned to position $i$ when $n$ customers are present. A symmetric discipline inserts an arrival into position $i$ with probability ${\gamma }_i(n+1)$. The same position rule for service and insertion makes transitions satisfy detailed balance for the word weights ${\prod }_i{\alpha }_{c_i}/{\mu }_{c_i}$. It accommodates class-dependent exponential service rates; processor sharing is an example.

A nonempty server divides its capacity equally among all customers present. With class-$r$ exponential service parameter ${\mu }_r$, population $n_r$, and total population $n$, the class departure rate is ${\mu }_r{n}_r/n$. The resulting class-count process is a continuous-time Markov chain.

A queue serves customers in arrival order. For multiple classes, retaining this order generally requires an ordered multiclass single-server queue state, rather than only a vector of class counts.