Simulating Random Signal Arrivals in a Communication System

Generate and visualize random samples of waiting times for multiple transmitters with different rates

In this example we'll analyze delays in signaling for systems in parallel and in series . First, waiting times at a receiver for signals coming from four independent transmitters are exponentially distributed with parameters

, and

, respectively.

Find the probability that the third transmitter's signal arrives first:

In[1]:=

Probability[Min[u,v,w,z]w,{uExponentialDistribution[λ1],vExponentialDistribution[λ2],wExponentialDistribution[λ3],zExponentialDistribution[λ4]}]

Out[1]=

λ3

λ1+λ2+λ3+λ4

Define the distribution of the earliest arriving signal (exponential with the combined rate

λ₁+λ₂+λ₃+λ₄

In[2]:=

dist=TransformedDistribution[Min[u,v,w,z],{uExponentialDistribution[λ1],vExponentialDistribution[λ2],wExponentialDistribution[λ3],zExponentialDistribution[λ4]}]

Out[2]=

ExponentialDistribution[λ1+λ2+λ3+λ4]

Find the average waiting time for the first signal at the receiver:

In[3]:=

Mean[dist]

Out[3]=

λ1+λ2+λ3+λ4

Simulates 100 random waiting times for the given parameters and plot them. The horizontal orange line marks the mean waiting time:

In[4]:=

Block[{λ1=1,λ2=0.5,λ3=0.7,λ4=0.3},ListPlot[{RandomVariate[dist,100],{{0,Mean[dist]},{100,Mean[dist]}}},Joined{False,True}]]

Out[4]=

Now, assume that the time delay in a logic element is exponentially distributed and that the average delay is

-9

seconds. The longest sequence of logic elements in a combinational logic network is six. Find the probability that delay through all six elements is longer than

8.5×

-9

seconds:

Find the exponential distribution parameter,

, that yields the given mean delay:

In[5]:=

Mean[ExponentialDistribution[λ]]Quantity[

-9

,"Seconds"]

Out[5]=



1000000000

In[6]:=

NSolve[%,λ]

Out[6]=

λ

1.×



Create a delay distribution for a chain of six independent logic elements, modeled as an Erlang distribution with shape = 6 and rate = 10⁹ Hz:

In[7]:=

logicDelayDistribution=(ErlangDistribution[6,λ]/.First[%])

Out[7]=

QuantityDistributionErlangDistribution[6,1.×

/Hz



Compute the probability that the total delay exceeds 8.5 nanoseconds:

In[8]:=

Probabilityd>Quantity[8.5

-9

,"Seconds"],dlogicDelayDistribution

Out[8]=

0.149597

Publisher Information

Contributed by: Wolfram Staff

Simulating Random Signal Arrivals in a Communication System

Related Symbols

Publisher Information