International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013
ISSN 2229-5518
Factors Reducing Call Blocking Probability by
Exploiting Erlang B &C Formulas
Hassan Ali, Hafiz Faiz Rasool, Ashiq Hussain
Abstract— Resource sharing among the telecom subscribers focuses on different occasions when the incoming request for some particular connection cannot be entertained immediately and this call request either have to wait or has to be rejected by the server. Traffic congestion in the server depends upon the statistical parameters of user behavior which varies with time. Statistical Equilibrium can solve the problem regarding traffic congestion in the server which implies that ''the probability per unit time of a new call request arising is equal to the probability per unit time of an existing call terminating so the average number of successful connections remains more or less constant''. But this is not practical approach, because user behavior regarding telecom traffic is random.This paper illustrates that what happened to new calls which arises at that time when server has fully occupied. What will be the probability for the rejection or acceptance of the new incoming call while hitting the server. When all servers are occupied then the new call will be rejected because n o path will be available to it. . The offered traffic and carried traffic are dependent on the design of our system and user behavior is independent of the system design as it is random in nature. The different assumption about the behavior of user produce significant information for both the probability distributions first is the number of occupied server and second is the loss traffic. Our work is to develop a model and to focus on the factors which will reduce the call blocking probability during overload traffic in switching system. This model is based on Erlang Formulas; we obtain call blocking probability for each type of traffic and by showing MATLAB results and mathematical models. We will examine the actual blocking occurs in system and present the new model which reduces the call blocking probability. sharing among the telecom subscribers focuses on different occasions when the incoming request for some particular connection cannot be entertained immediately and this call request either have to wait or have to be rejected by the server. Traffic congestion in the server depends upon the statistical parameters of user behavior which varies with time. Statistical Equilibrium can solve the problem regarding traffic congestion in the server which implies that ''the probability per unit time of a new call request arising is equal to the probability per unit time of an existing call terminating so the average number of successful connections remains more or less constant''.
Index Terms— Statistical Equilibrium, Traffic congestion, Molina equation, Erlang-B or Blocked Call Lost (BCL), Erlang-C or Blocked Call
Wait (BCW), Molina Equation or Blocked Call Held (BCH)
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witching systems undergoes many problems regarding telecom traffic. There are different conditions which affects
We know that when N approaches to infinity [1], then binomi- al distribution approaches to Poisson distribution.
the performance of the switching systems. When there is finite
K
N K
number of sources, the new incoming call offered to the sys-
tem can be affected by the calls which are currently in progress in the system, and when system is fully occupied,
N A
K N
1 A
N
K
A
K 1
e A
......... (1)
then new incoming call undergoes in blocking state which A
results in CONGESTION. When the number of sources are
This means that
N
approaches to zero as N approach-
large the probability of new call arising is independent of the
number of call in progress [1]. In the circuit switched systems when voice traffic is taken into account, incoming calls from the terminals are not statistically independent for 100% suc- cess rate because they experiences congestion when the server is being well occupied.
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Hassan Ali, Assistant Professor, Department of Electrical & Electronics
Engineering The Islamia university of Bahawalpur, Pakistan.
E-mail: nasasway@yahoo.com
Hafiz Faiz Rasool, Lecturer, Department of Electrical & Electronics
Engineering The Islamia university of Bahawalpur, Pakistan.
E-mail: engr.faiz07@gmail.com
Hafiz Ashiq Hussain, Associate Proffessor, Department of Electrical &
Electronics Engineering The Islamia university of Bahawalpur, Pakistan.
E-mail: hafizashiq@hotmail.com
es to infinity. It shows that probability of new call originating will be decreased significantly due to congestion [1].
According to congestion theory, the value of the of-
fered traffic exceeds then the rated value of the server which experiences congestion as no new call accepted due to the fact of the server is fully occupied. There are two ways of conges- tion for the blocking. first is TIME congestion, which is the percentage of time that all servers in a group are busy. Second is CALL or DEMAND congestion which is proportion of call arising which do not find a free server [2].
When the telecom traffic is overloaded due to the
large number of hits on the server, we consider three possibili- ties to assess that particular state. These possibilities are [1]
(a) Blocked Calls Held (b) Blocked Calls Lost (c) Blocked Calls Wait
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International Journal of Scientific & Engineering Research Volume 4, Issue 4, April -2013
ISSN 2229-5518
355
In BCH, the blocked call continuously demands for service for duration of the holding time, and wishes to get successful by hunting the available slot if in this duration the server be- comes free. But on the other hand this call will not be consi- dered as a 100% successful call because some or most of the part of that particular call request will be eliminated or lost due to the overlap of the other call requests during that time period. For example the Fig 1 below [3]
Fig 1. Blocked Calls Held [3]
Blocked calls are held by the system and serviced when the necessary facility become available, each arrival require ser- vice for a continuous period of time as shown in Fig 1 and terminates request independently when service is unavailable due to congestion. We use MOLINA equation to find held in block calls [4].
In Blocked Call Lost (BCL) scenario, the call request
hits the server in the particular time when the server is busy. So the call do not avail the service and the subscriber hangs up the call so call is cleared [5]. To find blocked calls lost we use Erlang-B formula that gives probability of blocked call lost in the system due to the unavailability of the server at any in- stant of time. For the clear understanding of BCL, Fig 1 can be taken into account again in which call 2 will be completely lost and cleared from the system, because it does not find slot at the specific time. It is because when call 2 hit the server, at that time call 1 was still in progress [6],[7],[8].
In blocked call wait (BCW) scenario, switching system
effective sources [1],[8],[9].
The GOS is called call congestion or lost probability and the blocking probabilities are called time congestion. Through Matlab Results, we find the statistical characteristics of blocking in switching system.
Our scope of work is the careful analysis of three scenarios, BCH, BCL and BCW by Matlab simulations and try to find out some specific traffic mathematical model, which can be the key factor to reduce the blocking probability of incoming call requests on the server. Researchers trying their level best to make such a system which may give an output of 100% suc- cessful call rate. But in practice this is not possible due to some factors like Transmission Medium, Server occupancy & its Design and Random User Behaviour regarding using the ser- vice during making their call requests.
When server is fully occupied then incoming call is blocked due to the server busy. So if call is held or lost then there will be loss of Erlangs. This will also reduce the efficiency of the system and will also reduce faith of consumer of such unrelia- ble service. This problem can be overcome by two ways.
1. If we use the scheme that the number of servers and users are equal then no blocking occurs. But this is so expensive and difficult to implement.
2. If the switching system has some mechanism through
which block call can wait until server becomes free. In this scheme there will be no or less loss of Erlangs.
Our research question is to identify and the cure of such fac- tors which plays main role for the failure of incoming call re- quests and block them not to enter in the system, which re- duces the efficiency of the system.
Moreover our research question is to find the cost at
which any call that hits the server when it is fully occupied and what will be the mathematical model by the help of which that specific call will not discarded from the system.
We have to visualize the Molina Equation, which that that '' When there are large number of sources when N approaches to infinity [1], then binomial distribution approaches to Pois- son distribution.
has some property through which blocked call can wait until
BT
A e A
server becomes free. The delay probability in blocked call wait
is calculated by Erlang-C formula. In delay system there are finite numbers of sources. Waiting calls are selected on queuing system and use the fact that first come first served or first in first out. The blocking probability and delay probabili- ty is based on queue size in comparison with the number of
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x M
( x 1)
...................... (2)
International Journal of Scientific & Engineering Research Volume 4, Issue 4, April -2013
ISSN 2229-5518
356
Where | 1. | Infinite number of sources |
A = the total traffic offered in units of erlangs | 2. | Calls arrivals are random |
N = the number of servers | 3. | Calls are served in order of arrival. |
During the Matlab simulation, we analyze the Molina equa- | 4. | Blocked calls are lost. |
tion with 20 call requests and with 0.5 erlang traffic intensity.
Fig 2 shows, in the start calls has highest value of be- ing served, when number of calls increases, then system starts showing business level and in this condition the availability of free server decreases as shown in figure. In result more re- quests will decrease the call success rate and BCH probability will be increased. Incoming will find no server or some por- tion of time in the server and will be held, blocked or served with too much lost information as shown in Fig 1, call 2 and call 3.
Fig 2. BCH (Molina Equation)
Call arises during the time when server is busy so the call do not avail the service and the subscriber hangs up the call so call is cleared. To find Block call lost we use ERLANG-B for- mula that gives probability of call lost in blocking of call for a system and given by [8],[10]
A N
B
N K
( N 1) ( )
During the Matlab simulation, we analyze Erlang-B equation
with 20 call requests and with 0.5 erlang traffic intensity.
We analyze the Fig 3 and found the abrupt change in the graph as the call requests increases on the server. We found that initially the blocking probability approaches to ze- ro and as soon as call requests hits the server, it gets busier and new call requests blocking probability increases. After 20 hits on the server, 21st hit will experience the blocking and that call will be cleared and will not be entertained by the sys- tem.
Fig 3. BCL (Erlang-B)
In BCW, Switching system exhibits a mechanism through which blocked calls can wait until server becomes free. In de- lay system there are finite numbers of sources. For the same queue, waiting calls are selected on queuing system and use the fact that first come first served or FIFO. For different queue system we use LIFO or SIRO (service in random order) mechanisms. The blocking probability and delay probability is based on queue size in comparison with the number of effec- tive sources. The delay probability in blocked call wait is cal- culated by Erlang-C formula [1],[11].
Where:
B=Blocking probability
K 0
(K 1)
.......................... (3)
η= A/M, Average Occupancy of Server
B= Blocking Probability.
N=Number of servers in system
A=Offered traffic
A= offered Traffic.
N=M= Number of servers in the system
Erlang-B formula also known as the Erlang loss formula is
AM 1
B
M 1 x M
/
formula use to find the blocking probability to describe the
probability of call which are loss in the system. This formula
tells that when circuit is fully occupied then new call will be
rejected and lost. Erlang B formula has following assumptions:
(M 1) (1 )
x 0 (x 1)
(M 1) (1 )
............ (4)
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International Journal of Scientific & Engineering Research Volume 4, Issue 4, April -2013
ISSN 2229-5518
357
When N servers are fully occupied and the call request hits the server, the delay systems performs its duty and places the call in queue unless and until the call finds free server [12].
The Erlang C formula tells the waiting probability in a
queuing system. Let’s we have k number of calls and N num- ber of servers. When K ≤ N then all the servers are not occu- pied and calls are not in waiting and when K > N then K - N calls will go in waiting scenario and will not dropped. This is
the main merit of Erlang-C formula.
when it hits on the fully occupied server while using Erlang- C
formula.
In result, I formulate different mathematical models and proposed a model for the very next call request P(o), which indicates clear reduced blocking probability for that specific call.
By visualizing LCC (lost calls cleared) system [12], we know that:
P( K )
1
(K 1)
A K P(o)
................... (5)
Where K= 1,2,3,------------------N. [12]
By the help of equation (5), I formulate the mathematical model for P(o), which is as follows.
Fig (4) Queuing Model [12]
N 1 x
( )
A N
x0
(x 1)
N A (N 1)
N A K 0 (K 1) ...... (6)
During the Matlab simulation, we analyze Erlang-C equation with 20 call requests. We analyze the Fig 5 and found that the
N A N 1 A x A N N K
blocking probability of call requests on the server when it is completely occupied are less than the blocking of call requests
N x 0 ( x 1)
1
( N 1)
K 0 (K 1) ......(7)
1
for Erlang-B formula.
N A N 1 A x A N N K
N x0 (x 1)
(N 1)
K 0 (K 1) ..........(8)
Then we know that:
P(0) 1
N K
Then,
P(o)
K 0 (K 1) ............................... (9)
1
( N A
N 1 x N
)
N x0 (x 1)
( N 1) ........... (10)
Fig (5) BCW (Erlang-C)
We found that on the usage of 0.5 Erlang by the calls, the serv- er exhibits less blocking probability for the requested calls in Erlang-C formula the in Erlang-B formula.
By the keen analysis of Molina equation, Erlang-B and Erlang- C formulas, it is observed that there are some key factors which behaves very important role for reducing the blocking probability of the requested calls on fully occupied servers. We analyzed these factors separately and developed an en- hanced mathematical model by the help of which we can un- derstand that what happens to the very next call request,
This is the required first call which hits the server when it is fully occupied.
Simulation results clearly shows in Fig 6, that when
p(o) hits the server while using Erlang-C mechanism, the serv- er does not clears the call but put it in queue and process it when there is free slot available. The graph also shows that reducing blocking probability for the call P(o) and for the suc- cessors of P(o) when they hits the fully occupied server.
During the MATLAB implementation, we analyze our pro- posed mathematical equation with 20 call requests.
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International Journal of Scientific & Engineering Research Volume 4, Issue 4, April -2013
ISSN 2229-5518
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Fig (6) Proposed Model for Call P(o)
After the keen analysis of situation in which the number of servers and users are equal then no blocking occurs. But this is so expensive and difficult to implement. From Matlab results we have find that Erlang C has less blocking probability than Erlang B because the calls are in wait system and they goes in queue if the server is fully occupied. When server is fully oc- cupied then incoming call is blocked due to the busy server. So if call is held or lost then there will be loss of Erlangs. This will also reduce the efficiency of the system and will also re- duce faith of consumer of such unreliable service.
We proposed a mathematical model, by the help of which that specific call will not discarded from the system in case of fully occupied server. We proposed the reduced call blocking factor such as P(0) which shows that when using Er- lang-C formula, the server does not clears the call but put it in queue and process it when there is free slot available. The proposed Mathematical Model is also defended with Matlab simulations.
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