Estimating Switchboard Capacity Using Probability Models

Model the likelihood of overload events using statistical distributions

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A switchboard gets about 100 calls each minute. Find the smallest capacity (in calls per minute) so that it overloads less than once every 60 minutes.

Define the probability that the number of received calls exceeds the switchboard capacity 𝑐 per minute:

In[1]:=

satprob=Probability[x>Quantity[c,"Minutes"],xQuantityDistribution[PoissonDistribution[100],"Minutes"],Assumptionsc∈Integers&&c>0]

Out[1]=

Gamma[1+c]-Gamma[1+c,100]

Gamma[1+c]

Plot the saturation probability for capacities between 110 and 140 calls per minute and compare it with the reliability threshold of

, shown as an orange horizontal line:

In[2]:=

DiscretePlot[satprob,{c,110,140},PlotRangeAll,Epilog{Orange,InfiniteLine[{0,1/60},{1,0}]},AxesLabel{"capacity","saturation probability"}]

Out[2]=

Find the smallest integer capacity 𝑐 that keeps the saturation probability below

In[3]:=

Minimize[{c,satprob≤1/60&&110<c<140},c,Integers]

Out[3]=

{122,{c122}}

Assume that the duration of telephone calls is exponentially distributed. The average length of a telephone call is 3.7 minutes. Find the probability that 9 consecutive phone calls will be longer than 25 minutes:

Find the exponential distribution parameter,

, that yields the given average call length:

In[4]:=

{sol}=NSolve[Mean[ExponentialDistribution[λ]]Quantity[3.7,"Minutes"],λ];

Out[4]=

λ

0.27027

per minute



Model the total duration of nine consecutive phone calls using an Erlang distribution with shape parameter 𝑘 = 9 and the previously computed rate

𝜆

In[5]:=

phoneDurationDistribution=(ErlangDistribution[9,λ]/.sol)

Out[5]=

QuantityDistributionErlangDistribution[9,16.2162],



Compute the probability that nine consecutive calls last longer than 25 minutes in total:

In[6]:=

Probability[d>Quantity[25,"Minutes"],dphoneDurationDistribution]

Out[6]=

0.760209

Related Symbols

PoissonDistribution
ExponentialDistribution
ErlangDistribution
QuantityDistribution
Probability
Minimize
NSolve
Plot
DiscretePlot

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Contributed by: Wolfram Staff