Tool Crib Simulation with Priority Classes

The problem

Mechanics come to a crib for tools. Class 1 mechanics arrive every 420 ± 360 seconds and keep the clerk busy for 300 ± 90. Class 2 mechanics arrive every 360 ± 240 seconds with quick requests of 100 ± 30. All the times are uniform over those ranges.

The clerk serves class 2 first. Nobody is interrupted: a class 1 request already under way is finished, but whenever the clerk frees up, a waiting class 2 mechanic goes ahead of every class 1 mechanic in line. The run is one eight-hour shift, 28,800 seconds.

Add up the averages and the clerk has 300/420 + 100/360 of an hour’s work in every hour, about 99%. The crib is close to full, and that is what makes the priority rule interesting.

Deck and model

In the deck, priority is the fifth field of GENERATE: ,,,1 and ,,,2. Each class has its own line so the statistics come out separately, and one SEIZE CLERK serves both, highest priority first.

chiaha-SLX says the same thing on the arrivals statement, priority = 2, and the clerk’s waiting line is kept in priority order. The rest of each mechanic’s actions are the deck’s blocks in the deck’s order.

GENERATE 420,360,,,1 QUEUE LINE1 SEIZE CLERK DEPART LINE1 ADVANCE 300,90 RELEASE CLERK TERMINATE GENERATE 360,240,,,2 QUEUE LINE2 SEIZE CLERK DEPART LINE2 ADVANCE 100,30 RELEASE CLERK TERMINATE GENERATE 28800 TERMINATE 1

import <h4>

module tool_crib_h4
     {
     facility  Clerk;
     queue     Line1, Line2;

     rn_stream Arrive1 seed=100000;
     rn_stream Arrive2 seed=200000;
     rn_stream Serve1  seed=300000;
     rn_stream Serve2  seed=400000;

     float     shift = 28800;
     boolean   poisson = FALSE;
     int       arrived1, arrived2;
     int       departed1, departed2;
     float     waited1, waited2;
     float     busy;

     object mechanic1
          {
          float arrived, service;
          actions
               {
               arrived = time;
               arrived1++;
               enqueue   Line1;
               seize     Clerk;
               depart    Line1;
               waited1 += time - arrived;
               departed1++;
               service = rv_uniform(Serve1, 210.0, 390.0);
               advance   service;
               busy += service;
               release   Clerk;
               terminate;
               }
          }

     // mechanic2 is the same with Line2, Serve2 and 70 to 130 seconds

     procedure main
          {
          if (poisson)
               {
               arrivals: mechanic1 iat = rv_expo(Arrive1, 600.0) priority = 1;
               arrivals: mechanic2 iat = rv_expo(Arrive2, 400.0) priority = 2;
               }
          else
               {
               arrivals: mechanic1 iat = rv_uniform(Arrive1, 60.0, 780.0) priority = 1;
               arrivals: mechanic2 iat = rv_uniform(Arrive2, 120.0, 600.0) priority = 2;
               }
          advance shift;
          report(system);
          }
     }

The workspace example deck/tool-crib, with the second mechanic class shortened to a comment here. The poisson switch swaps in exponential arrivals.

One shift, twenty times

Class 2 hardly notices the crowd. Class 1 absorbs all of it, and its wait swings by a factor of ten from one shift to the next.

Measure	Mean of 20 runs	Lowest to highest run
Class 1 mechanics served	66.1 ± 1.1	61 to 70
Class 2 mechanics served	80.0 ± 1.7	74 to 88
Class 1 mean wait	954.0 s ± 306.0	209.2 s to 2,523.1 s
Class 2 mean wait	140.6 s ± 6.2	109.0 s to 170.2 s
Clerk utilization	95.30% ± 1.20	90.84% to 99.17%

± is the 95% confidence interval of the mean over 20 runs. There is no long-run row for the deck as written: with the clerk this close to full, class 1’s line never settles, and a longer run only widens the spread.

A version with an exact answer

Turn on poisson = TRUE and arrivals become exponential, every 600 and 400 seconds on average. The clerk drops to 75% busy and the waits follow Cobham’s formulas for a non-preemptive priority queue. That version, run for two million seconds, is the fourth case on the verification page, and the simulated waits for both classes sit inside their intervals.

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