Communication Math

Khaled Mahmud

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Erlang B

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Erlang B Formula

Given,
N = Total number of truncked channels, Erl
A = Total traffic intensity, Erl

Probability of call blocking, AKA Grade of Service (GoS), is determined by ErlangB formula.

\begin{equation} GoS = \frac{\cfrac{A^N}{N!}}{\sum_{i=1}^N \cfrac{A^i}{i!}} \end{equation}

#Erlang B formula
from math import factorial
def ErlangB(A,N):
    '''
    Calculate Probability of call blocking (Grade of Service)
    Input: 
        A- Total offered load (Busy Hour Traffic (BHT))
        N- Number channels
    '''
    ratiosum=0
    for i in range(N+1):
        ratio=(A**i)/factorial(i)
        ratiosum+=ratio
    return (ratio/ratiosum)

#Plotting GoS familty of curves, derived from ErlangB formula
import numpy as np
import matplotlib.pyplot as plt

#Create x axis--> Offered Load values
x=np.linspace(0.1,0.9,19)
x=np.concatenate([x,10*x,100*x,[100]])
aarray=list(x) #28 items 
#Create Channel number list
carray=[1,2,3,4,5,6,7,8,9,10,12,14,16,18,20,25,30,35,40,45,50,60,70,80,90,100] #26 items

prlist=[]
prsetlist=[]

for c in carray:
    for a in aarray:
        pr=ErlangB(a,c)
        prlist.append(pr)
    prsetlist.append(prlist)
    prlist=[]

#Plot in log-log scale

plt.figure(1, figsize=(16,10))
for prsetlistitem in prsetlist:
    plt.loglog(aarray, prsetlistitem)

plt.grid(which='both', axis='both')
plt.xlim(0.1,100)
plt.ylim(0.001, 0.1)
plt.xlabel('Traffic Intensity in Erlang')
plt.ylabel('Probability of Blocking')
plt.title ('Number of Trunked Channels (N)')
plt.xscale('log')
plt.show()

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Sample Trunking Problem

With 60 avaiable truncked channels, what will be the probalibility of blocking (GoS) if the offered load is 50 erlang?

N=60
A=50

pr=ErlangB(A, N)
print("Probability of blocking: {:.3f} ".format(pr))
print("Or GoS: {:.2}% ".format(pr*100))

Probability of blocking: 0.022 
Or GoS: 2.2% 

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Erlang C

Given the traffic intensity, A and number of agent, N, the probability of a caller waiting is given by Erlang C formula.

\begin{equation} P_{w}=\frac{\frac{A^{N}}{N!}\frac{N}{N-A}}{\left ( \sum_{i=0}^{N-1}\frac{A^{i}}{i!} \right )+\frac{A^{N}}{N!}\frac{N}{N-A}} \end{equation}

#Erlang C formula
from math import factorial
def ErlangC(A,N):
    '''
    Calculate Probability of call waiting 
    Input: 
        A- Total offered load (Busy Hour Traffic (BHT))
        N- Number channels (servers, agents)
    '''
    ratiosum=0
    for i in range(0,N):
        ratio=(A**i)/factorial(i)
        ratiosum+=ratio
    ratio=(A**N)/factorial(N)
    ratio2=N/(N-A)*ratio
#    print (ratio, ratiosum)
    return (ratio2/(ratiosum+ratio2))

#Example
N=11
A=10

pr=ErlangC(A, N)
print("Probability of waiting: {:.2f}% ".format(pr*100))

2505.210838544172 12842.305114638448
Probability of waiting: 68.21% 

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