Probability / Blocking

When systems operate under finite physical capacity (queue limits, buffer bounds, or restricted connections), requests that arrive when the system is full experience immediate degradation. The blocking and loss probability metric quantifies how often the system reaches capacity saturation and rejects incoming work.

The Core Expression

The percentage probability of finding the system with zero available capacity is expressed directly as:

P{#Slots = 0} * 100

The operator P{ } calculates the fraction of simulation time during which the condition was true. Multiplied by 100, it yields the percentage of time the system operated at maximum saturation.

Blocking vs. Dropping: Modeling Distinction

Two distinct behaviors occur when a system reaches capacity:

Property	Blocking	Dropping / Loss
Behavior	Arriving items wait outside until a slot is freed	Arriving items that find no slots are discarded immediately
Real-world example	Assembly line conveyor belt that pauses	Dropped phone call, discarded UDP packet
Petri Net Modeling	Arrival transition consumes a slot token to fire	Arrival transition always fires and branches into Accept or Drop

Modeling Explicit Workload Dropping

When rejected requests are discarded, the net models the branching logic explicitly using immediate transitions and inhibitor arcs:

json

{
  "modelName": "Fila com recusa de clientes",
  "definitions": [
    {
      "name": "TEMPO_ENTRE_CHEGADAS",
      "type": "DOUBLE",
      "value": "10"
    },
    {
      "name": "TEMPO_DE_ATENDIMENTO",
      "type": "DOUBLE",
      "value": "5"
    },
    {
      "name": "VAGAS",
      "type": "INTEGER",
      "value": "3"
    },
    {
      "name": "PASSAGEM",
      "type": "DOUBLE",
      "value": "0.0001"
    }
  ],
  "places": [
    {
      "name": "Vagas",
      "initialMarking": 3,
      "stringMarking": "VAGAS",
      "initialStringMarking": "VAGAS"
    },
    {
      "name": "Fila",
      "initialMarking": 0,
      "stringMarking": "0",
      "initialStringMarking": "0"
    },
    {
      "name": "Atendente",
      "initialMarking": 1,
      "stringMarking": "1",
      "initialStringMarking": "1"
    },
    {
      "name": "EmAtendimento",
      "initialMarking": 0,
      "stringMarking": "0",
      "initialStringMarking": "0"
    },
    {
      "name": "Chegando",
      "initialMarking": 0,
      "stringMarking": "0",
      "initialStringMarking": "0"
    },
    {
      "name": "Perdidos",
      "initialMarking": 0,
      "stringMarking": "0",
      "initialStringMarking": "0"
    }
  ],
  "rewardMeasures": [
    {
      "name": "cheio_pct",
      "expression": "P{#Vagas = 0} * 100"
    },
    {
      "name": "taxa_de_perda",
      "expression": "E{#Perdidos} / PASSAGEM"
    },
    {
      "name": "vazao",
      "expression": "E{#EmAtendimento} / TEMPO_DE_ATENDIMENTO"
    }
  ],
  "transitions": [
    {
      "name": "Chegada",
      "type": "TIMED",
      "firingPolicy": "SINGLE_SERVER",
      "raceType": "RACE_WITH_ENABLING_MEMORY",
      "priority": 1,
      "distribution": {
        "type": "exponential",
        "parameters": {
          "Mean delay": "TEMPO_ENTRE_CHEGADAS"
        }
      },
      "inputArcs": [],
      "outputArcs": [
        {
          "type": "OUTPUT",
          "place": "Chegando",
          "multiplicity": 1
        }
      ],
      "inhibitorArcs": []
    },
    {
      "name": "Inicio",
      "type": "IMMEDIATE",
      "firingPolicy": "SINGLE_SERVER",
      "raceType": "RACE_WITH_ENABLING_MEMORY",
      "priority": 1,
      "weight": 1,
      "inputArcs": [
        {
          "type": "INPUT",
          "place": "Fila",
          "multiplicity": 1
        },
        {
          "type": "INPUT",
          "place": "Atendente",
          "multiplicity": 1
        }
      ],
      "outputArcs": [
        {
          "type": "OUTPUT",
          "place": "EmAtendimento",
          "multiplicity": 1
        }
      ],
      "inhibitorArcs": []
    },
    {
      "name": "Fim",
      "type": "TIMED",
      "firingPolicy": "INFINITY_SERVER",
      "raceType": "RACE_WITH_ENABLING_MEMORY",
      "priority": 1,
      "distribution": {
        "type": "exponential",
        "parameters": {
          "Mean delay": "TEMPO_DE_ATENDIMENTO"
        }
      },
      "inputArcs": [
        {
          "type": "INPUT",
          "place": "EmAtendimento",
          "multiplicity": 1
        }
      ],
      "outputArcs": [
        {
          "type": "OUTPUT",
          "place": "Atendente",
          "multiplicity": 1
        },
        {
          "type": "OUTPUT",
          "place": "Vagas",
          "multiplicity": 1
        }
      ],
      "inhibitorArcs": []
    },
    {
      "name": "Aceita",
      "type": "IMMEDIATE",
      "firingPolicy": "SINGLE_SERVER",
      "raceType": "RACE_WITH_ENABLING_MEMORY",
      "priority": 1,
      "weight": 1,
      "inputArcs": [
        {
          "type": "INPUT",
          "place": "Chegando",
          "multiplicity": 1
        },
        {
          "type": "INPUT",
          "place": "Vagas",
          "multiplicity": 1
        }
      ],
      "outputArcs": [
        {
          "type": "OUTPUT",
          "place": "Fila",
          "multiplicity": 1
        }
      ],
      "inhibitorArcs": []
    },
    {
      "name": "Recusa",
      "type": "IMMEDIATE",
      "firingPolicy": "SINGLE_SERVER",
      "raceType": "RACE_WITH_ENABLING_MEMORY",
      "priority": 1,
      "weight": 1,
      "inputArcs": [
        {
          "type": "INPUT",
          "place": "Chegando",
          "multiplicity": 1
        }
      ],
      "outputArcs": [
        {
          "type": "OUTPUT",
          "place": "Perdidos",
          "multiplicity": 1
        }
      ],
      "inhibitorArcs": [
        {
          "type": "INHIBITOR",
          "place": "Vagas",
          "multiplicity": 1
        }
      ]
    },
    {
      "name": "Descarta",
      "type": "TIMED",
      "firingPolicy": "INFINITY_SERVER",
      "raceType": "RACE_WITH_ENABLING_MEMORY",
      "priority": 1,
      "distribution": {
        "type": "exponential",
        "parameters": {
          "Mean delay": "PASSAGEM"
        }
      },
      "inputArcs": [
        {
          "type": "INPUT",
          "place": "Perdidos",
          "multiplicity": 1
        }
      ],
      "outputArcs": [],
      "inhibitorArcs": []
    }
  ]
}

Accept Path: Consumes 1 available slot token and routes the request into the waiting queue.
Drop Path: Connects to Slots via an inhibitor arc. It fires only when Slots has 0 tokens, routing the request to Dropped.

An immediate transition Discard flushes tokens from Dropped, enabling direct computation of the drop rate:

E{#Dropped} / TRANSIT_DELAY

Verifying Flow Conservation

In any properly formulated model with dropping, total arrival rate must balance against served throughput and loss rate:

Quantity	Simulation Output
Nominal arrival rate	0.1000 items/min
Served throughput	0.0933 items/min
Loss rate	0.0067 items/min
Served throughput + Loss rate	0.1000 items/min ✓

If this sum diverges, an arc in the model is creating or destroying tokens unexpectedly.

Correlating Blocking with Response Time

Restricting queue capacity reduces waiting time for admitted users, but at the cost of rejecting others:

Available slots	% Time full	Completed throughput	Mean response time
1 slot	33.4%	0.067	5.0 min
3 slots	6.7%	0.093	7.9 min
10 slots	0.0%	0.100	10.0 min

With 1 slot, the average response time is the fastest in the table (5 minutes) — but the system turns away 1 out of every 3 incoming requests!

Never present response time in isolation

Presenting lower response times without showing the corresponding rejection rate masks severe service degradation. A fast response time achieved by dropping significant traffic indicates misconfigured capacity.

Next steps

Learn how to configure and execute stochastic engines in 5 · Execution and Analysis.

Probability / Blocking ​

The Core Expression ​

Blocking vs. Dropping: Modeling Distinction ​

Modeling Explicit Workload Dropping ​

Verifying Flow Conservation ​

Correlating Blocking with Response Time ​