Designation Of Queue And Symbols Used In Queuing Models

Introduction:

In queuing model, two basic constituents are considered i.e. arrival rate and service rate; these two are the main problems of waiting line. For such type of problems is expressed in different forms and notation.

 Designation:A queue is designated or described as shown below: A model is expressed as

A/B/S: (d / f) where,

A: Arrival pattern of the units, given by the probability distribution of inter - arrival time of units.

 For example, Poisson distribution, Erlang distribution, and inter arrival time is 1 minute or 10 units arrive in 30 minutes etc.

B: The probability distribution of service time of individual being actually served.

 For example the service time follows negative exponential distribution and 10 units are served in 10 minutes or the service time is 3 minutes, etc.

S: The number of service channels in the system.

For example the item is served at one service facility or the person will receive service at 3 facilities etc.

d: Capacity of the system. That is the maximum number of units the system can accommodate at any time.

 For example, the system has limited capacity of 40 units or the system has infinite capacity etc.

f: The manner or order in which the arriving units are taken into service i.e. FIFO / LIFO / SIRO /Priority.

Notation

X: Inter arrival time between two successive customers (arrivals).

Y: The service time required by any customer.

w: The waiting time for any customer before it is taken into service.

v: Time spent by the customer in the system.

n: Number of customers in the system, that is customers in the waiting line at any time,

including the number of customers being served.

Pn (t): Probability that .n. customers arrive in the system in time .t..

Φn (t): Probability that .n. units are served in time .t..

U (T): Probability distribution of inter arrival time P (t ≤ T).

V (T): Probability distribution of servicing time P (t ≤ T).

F (N): Probability distribution of queue length at any time P (N ≤ n).

En: Some state of the system at a time when there are .n. units in the system.

λn: Average number of customers arriving per unit of time, when there are already .n. units in the system.

λ: Average number of customers arriving per unit of time.

μn: Average number of customers being served per unit of time when there are already .n. units in the system.

μ:Average number of customers being served per unit of time.

1 / λ : Inter arrival time between two arrivals.

1 / μ : Service time between two units or customers.

ρ = (λ /μ) :System utility or traffic intensity which tells us how much time the system was utilized in a given time.

 For example given time is 8 hours and ifρ = 3 / 8,  it means to say that out of 8 hours the system is used for 3 hours and (8 . 3 = 5) 5 hours the is idle.