Simulation | Monte Carlo Method | Mass Servicing System

Question: Consider one of the simplest systems of mass servicing. This system consists of \(n\) lines (or channels, or servicing stations) each of which can "serve on the customers". The system receives requests arriving at random moments of time. Each request arrives at the \(N_{1}\) line. If the arrival time of the k-th request ( let us call it \(T_{k}\) ) finds this line free, the line starts servicing the request; this takes \(t_{b}\) minutes (\(t_{b}\) is the holding time of the line). If \(N_{1}\) line is busy at the moment \(T_{k}\) the request is immediately transferred to the \(N_{2}\) line and ... so on. Finally, if all n lines are busy at the moment \(T_{k}\) , the system rejects the request. The problem is, what will be the (average) number of requests serviced by the system during the period \(T\) and how many rejections will be given?

Simulation

• Firstly, it is a very long list of images (several pages long)
• Each image has a label on the top of it showing
  • \(\lambda_{a}\) the rate of arrival rate of the customers and the times of the arrival are exponentially distributed
  • \(\lambda_{s}\) the rate of service for the customers and the service times are exponentially distributed as well
• The chart for each of the server has as blue line, which is binary in nature
  • the server is busy when the blue line is above the \(x\) axis and
  • the server is free when the blue line is on the \(x\) axis
• As you scroll down the images, you will see that the pattern of the images is changing as in, which server is busy over the lifetime of the experiment
• Notice how the "busyness" of the servers that are in sequence is highly dependent on the rate of the service
  • when the rate of the service is comparable or lesser than the rate of the arrival, the servers at the end also get some tasks as the first ones are taking a long time to finish what they have started and the servers next in line are free to server the incoming customers
  • when the rate of the service is very high compared the to the rate of arrival, the servers at the beginning of the queue are busy for most of the lifetime and the ones at the end do not even get to see any customers

Simulation Images

Below is a slide show of the various combinations of the \(\lambda_{a}\) and \(\lambda_{s}\)

• Rate of arrivals, \(\lambda_{a} = \frac{5.0}{hour}\)
• Rate of service, \(\lambda_{s} = \frac{k}{hour} \mid k \in \{0.1,0.2,...,5.0\}\)
• Observations to be made
• The tasks are passed down server \(1\)
• When server \(1\) can no longer handle any requests, the request is passed on to server \(2\)
• This is continued until all the servers can no longer accept any requests
• Observations to be made from the animation below
• As the \(\lambda_a\) is constant
• When \(\lambda_s\) is low, almost all servers are busy
• As the \(\lambda_s\) is increasing, the load on the later servers is almost nothing

Code

ClearAll[status, currentStatus];
status[list_List, time_] := Boole[Or @@ (Min[#] < time <= Max[#] & /@ list)]
currentStatus[list_List, time_] := ! Min[#] < time <= Max[#] &[Last[list]]

Module[{servers = 10, serverStatus, nCustomers = 120,
  arrivals, \[Lambda]a = 5, \[Lambda]s = #, firstFree, plot},
  arrivals = Accumulate[RandomVariate[ExponentialDistribution[\[Lambda]a], nCustomers]];
  serverStatus = ConstantArray[{{0, 0}}, servers];

  Quiet@Table[
    firstFree =
        First[Position[
          currentStatus[#, arrivals[[r]]] & /@ serverStatus,
          True]][[1]];
    If[firstFree =!= Null,
      firstFree = First[firstFree][[1]];
      serverStatus[[firstFree]] =
          serverStatus[[firstFree]] ~
              Join ~ {{arrivals[[r]],
            arrivals[[r]] +
                RandomVariate[ExponentialDistribution[\[Lambda]s]]}};,
      Print["No server free"]]
    , {r, 1, Length@arrivals}];

  plot =
      Labeled[Framed@
          Column[MapThread[
            Plot[status[#1, r], {r, 1, 100}, Exclusions -> None,
              ExclusionsStyle -> Red, ImageSize -> 788,
              AspectRatio -> 0.1, Frame -> True,
              PlotRange -> {{0, Max@arrivals}, {0, 1.1}},
              PlotLabel -> "Server " <> #2, FrameTicks -> False,
              FrameLabel -> {None,
                Style["NotBusy-Busy", Bold]}] &, {serverStatus,
              ToString /@ Range[Length@serverStatus]}]],
        "\!\(\*SubscriptBox[\(\[Lambda]\), \(a\)]\) = " <>
            ToString[\[Lambda]a] <>
            " : \!\(\*SubscriptBox[\(\[Lambda]\), \(s\)]\) = " <>
            ToString[\[Lambda]s], Top];

  Export[
    StringReplace[NotebookFileName[],
      ".nb" ->
          "_rates_" <> ToString[\[Lambda]a] <> "_" <>
              ToString[\[Lambda]s] <> ".png"], plot, ImageSize -> 788,
    ImageResolution -> 1000]
] & /@ Range[0.1, 5, 0.1]