The service of incoming calls depends on the number of lines. If number of lines equal to the number of subscribers, there is no question of traffic analysis. But it is not only uneconomical but not possible also. So, if the incoming calls finds all available lines busy, the call is said to be blocked. The blocked calls can be handled in two ways.
The type of system by which a blocked call is simply refused and is lost is called loss system. Most notably, traditional analog telephone systems simply block calls from entering the system, if no line available. Modern telephone networks can statistically multiplex calls or even packetize for lower blocking at the cost of delay. In the case of data networks, if dedicated buffer and lines are not available, they block calls from entering the system.
In the second type of system, a blocked call remains in the system and waits for a free line. This type of system is known as delay system. In this section loss system is described. Delay system is discussed in the section 8.5.
These two types differs in network, way of obtaining solution for the problem and GOS. For loss system, the GOS is probability of blocking. For delay system, GOS is the probability of waiting.
Erlang determined the GOS of loss systems having N trunks, with offered traffic A, with the following assumptions. (a) Pure chance traffic (b) Statistical equilibrium (c) Full availablity and (d) Calls which encounter congestion are lost. The first two are explained in previous section. A system with a collection of lines is said to be a fully-accessible system, if all the lines are equally accessible to all in arriving calls. For example, the trunk lines for inter office calls are fully accessible lines. The lost call assumption implies that any attempted call which encounters congestion is immediately cleared from the system. In such a case, the user may try again and it may cause more traffic during busy hour.
The Erlang loss system may be defined by the following specifications.
1. The arrival process of calls is assumed to be Poisson with a rate of X calls per hour.
2. The holding times are assumed to be mutually independent and identically distributed random variables following an exponential distribution with 1/|u seconds.
3. Calls are served in the order of arrival.
There are three models of loss systems. They are :
1. Lost calls cleared (LCC)
2. Lost calls returned (LCR)
3. Lost calls held (LCH)
All the three models are described in this section. 8.4.1. Lost Calls Cleared (LCC) System
The LCC model assumes that, the subscriber who does not avail the service, hangs up the call, and tries later. The next attempt is assumed as a new call. Hence, the call is said to be cleared. This also referred as blocked calls lost assumption. The first person to account fully and accurately for the effect of cleared calls in the calculation of blocking probabilities was A.K. Erlang in 1917.
Consider the Erlang loss system with N fully accessible lines and exponential holding times. The Erlang loss system can be modeled by birth and death process with birth and death rate as follows.
Substituting equation (8.48 and 8.49) in the above equation, we get k
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