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State queue notation, Queue discipline and Queue behavior.

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Queuing Notation:

Kendall suggested a notational system to describe a queuing system.

Following conventions are based on format A/B/C/N/K

A: Interarrival time distribution or arrival process commonly used shorthands are

• m (markovian, memoryless, exponential)
• D (constant, Deterministic)
• $E_A$ (Erlang of order k)
• G (General or arbitary)
• $G_I$ (General and independent)
• H (Hyper exponential)
• PH (Phase - Type)

B: Service time distribution

C: No. of parallel server

• 1, multiple, infinite

N: System capacity or queue size

• Finite, infinite

K: Size of calling population

• Finite, infinite

When N and K are infinite, they can be dropped from the notation

Few queuing notations are terminologies for parallel server system are given in the table.

Notation Meaning
$P_n$ Steady state probability of having 'n' customers in s/m
$\lambda$ Arrival Rate
$\mu$ Service rate of a server
$\rho$ Server utilization
L Long run time avg no. of customers in s/m
$L_Q$ Long run time avg no. of customers in queue
$\omega$ Long run avg time spent in s/m per customers
$\omega_\alpha$ Long run avg time spent in queue per customers

Other queuing notations are:

m / a / 1 queue: It has a single server with interarrival time which is exponentially distributed and service time is general with mean $V_\mu$ and variance $\sigma^2$

$P_0$: Steady state probablity of having zero customers in system

m / m / 1 queue: It has a single server with exponentially distributed interarrival time with mean $Y_\lambda$. The service time is also exponentially distributed with mean $Y_\mu$

Queue Discipline

• It is logical ordering of customers in queue or also called as service principle.

• It is the manner by which customer will be chosen for service when a server becomes free and queue has formed.

• Properties of a queuing system which are concerned with waiting times, depend on queue disciple.

• Some of them are ICIS / FIFO, LIFO, SIRO, SPT, RR (Round Robin), Priority

• Normally when human beings are involved and service provider is also human being, ICFS queue discipline is followed. When human beings are not part of the system, queue discipline can vary or change.

Queue Behavior

It is customer action while in queue waiting for service. Customers may balk, renege or jockey based on status of the queue.

Balk: Leave when they see line is too long

Renege: Leave after being in line when they see that line is moving too slowly.

Jockey: Move from one line to another if they think they have chosen a slow line.