DiscreteMarkovProcess
StationaryDistribution
RandomFunction

Mathematics
System Modeling

Modeling a Taxi Service as a Markov Chain

Model a taxi service through a discrete Markov process

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A taxi can travel between three locations: the airport, the suburbs, and the city. From the city, the next trip is to the airport or the suburbs each with probability 1/4, and to somewhere else in the city with probability 1/2. From the suburbs, the next trip is to the city with probability 7/10, to somewhere else in the suburbs with probability 1/20, and to the airport with probability 1/4. From the airport, the next trip is to the city with probability 7/10 and to the suburbs with probability 3/10.

Model the taxi using a discrete Markov process with state 1 representing the city, state 2 representing the suburbs, and state 3 representing the airport. Assume the taxi starts at the airport:

In[1]:=

taxi=DiscreteMarkovProcess[3,{{1/2,1/4,1/4},{7/10,1/20,1/4},{7/10,3/10,0}}];

Visualize the process as a graph:

In[2]:=

With{panelLabelWrapper=Panel[#,FrameMargins0,BackgroundLighter[Pink,0.8]]&},Graphtaxi,VertexLabels{1Placed["City",Center],2Placed["Suburbs",Center],3Placed["Airport",Center]},





Out[2]=

Simulate a typical sequence of taxi runs:

In[3]:=

timeline=RandomFunction[taxi,{0,50}]

Out[3]=

TemporalData

Time: 0 to 50
Data points: 51	Paths: 1



Plot the results:

In[4]:=

ListPlot[timeline,Ticks{Automatic,{{1,"City"},{2,"Suburbs"},{3,"Airport"}}},FillingAxis]

Out[4]=

Find the stationary distribution:

In[5]:=

dist=StationaryDistribution[taxi]

Out[5]=

ProbabilityDistribution

Boole[x.1]+

Boole[x.2]+

Boole[x.3],{x.,1,3,1}

The stationary distribution indicates that the taxi spends

of its time in the city:

In[6]:=

Options

In[7]:=

DiscretePlot[PDF[dist,x],{x,{1,2,3}},ExtentSizeFull,PlotRange{{0.5,3.5},{0,1}},Ticks{{1,2,3},Automatic}]

Out[7]=

External Links

https://www.wolframcloud.com/obj/online-courses/introduction-to-finite-mathematics/regular-markov-chains.html
https://reference.wolfram.com/language/guide/MarkovProcesses.html

Related Symbols

DiscreteMarkovProcess
StationaryDistribution
RandomFunction

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Contributed by: Wolfram Staff

Wolfram Language Example Repository

Modeling a Taxi Service as a Markov Chain

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