DELAY.MOD

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Contents

Description

DELAY.MOD represents a single-server queue that keeps track of how many customers have waited for more than L time units. This model can be used to estimate the probability that a customer waits less than a specified time (see comments), and to infer Little's Law using event counts.

State Variables

State Variables in DELAY.MOD
Variable Name Abbreviation Variable Description Size Type
QUEUE Q Number of jobs in queue 1 Integer
RESOURCES R Number of idle resources in model 1 Integer
S S Count of START Events 1 Integer
D D Count of DELAY Events 1 Integer
W W Proportion of jobs waiting longer than L 1 Real
L L Waiting time limit - to estimate Prob{Wait<L} 1 Real
PROB PROB Estimate of Prob{Wait<L} 1 Real

Vertices

Vertices in DELAY.MOD
Vertex Name Vertex Description State Changes
RUN The simulation is started None
ARRIVE Customers arrive in the queue Q=Q+1
START Jobs start service Q=Q-1, R=R-1, S=S+1
FINISH Jobs finish processing R=R+1
DELAY Counts delay events and number of customers who wait longer than L time units D=D+1, W=W+(S>=D), PROB=W/D

Initialization Conditions

Initialization Conditions in DELAY.MOD
Variable Description
RESOURCES Initial number of resources
L Waiting time limit

Event Relationship Graph

DELAY.MOD
DELAY.MOD

English Translation

An English translation is a verbal description of a model, automatically generated by SIGMA.

The SIGMA Model, DELAY.MOD, is a discrete event simulation. 
It models estimates the probability that the Wait<L.
I. STATE VARIABLE DEFINITIONS.
For this simulation, the following state variables are defined:
R: Number of idle resources in model   (integer valued)
Q: Number of jobs in queue   (integer valued)
S: Count of Start Events   (integer valued)
D: Count of Delay Events   (integer valued)
W: Proportion of jobs waiting longer than L  (real valued)
L: Waiting time limit - to estimate Prob{Wait<L}  (real valued)
PROB: Estimate of Prob{Wait<L}  (real valued)
II. EVENT DEFINITIONS.
The SIGMA Model, DELAY.MOD, is a discrete event simulation. It models estimates the probability
Simulation state changes are represented by event vertices (nodes or balls) in a SIGMA graph.  
Event vertex parameters, if any, are given in parentheses.  
Logical and dynamic relationships between pairs of events are represented in a SIGMA graph 
by edges (arrows) between event vertices.  
Unless otherwise stated, vertex execution priorities, to break time ties, are equal to 5.
1. The Run(R,L) event occurs when the run is started.  
   Initial values for, R,L, are needed for each run.  
   After every occurrence of the Run event:  
   Unconditionally, schedule the first job arrival; 
   that is, schedule the Arrive() event to occur without delay.
2. The Arrive() event occurs when customers arrive in the queue.
   This event causes the following state change(s):
   Q=Q+1
   After every occurrence of the Arrive event:
   Unconditionally, schedule the next job arrival; 
   that is, schedule the Arrive() event to occur in (1/.9)*ERL{1} time units.  
   If R>0, then there is an idle resource; 
   that is, schedule the Start() event to occur without delay.
   Unconditionally, a job arrived L time units ago; 
   that is, schedule the Delay() event to occur in L time units.  
   (Time ties are broken by an execution priority of 6.)
3. The Start() event occurs when jobs start service.
   This event causes the following state change(s):
   Q=Q-1
   R=R-1
   S=S+1
   After every occurrence of the Start event:
   Unconditionally, the job will finish after its processing time; 
   that is, schedule the Finish() event to occur in 1*ERL{1} time units.
4. The Finish() event occurs when jobs finish processing.
   This event causes the following state change(s):
   R=R+1
   After every occurrence of the Finish event:
   If Q>0, then there are jobs waiting, start on the next job; 
   that is, schedule the Start() event to occur without delay.
5. The Delay() event:
   This event causes the following state change(s):
   D=D+1
   W=W+(S>=D)
   PROB=W/D
   No additional events are scheduled here.

Comments

Estimating the probability that a customer waits less than a specific time using DELAY.MOD:

Suppose that we wish to compute the following probability distribution:

W is the average time a job waits in the queue. We can easily do this without storing any information for first-come-first-served (FIFO) queues. This is done by adding the DELAY event (see the Event Relationship Graph) that is scheduled to occur τ time units after each ARRIVAL event. The DELAY event simply tells us that a job arrived τ minutes ago. If service has not started by that time, then the customer has waited in queue longer than τ. We keep a count of the DELAY events that occur by time t, D(t). Whenever a START event occurs and D(t)≥S(t), then that job has waited in line at least τ. Let W be the total number of START events where D(t)<S(t). Our estimator of FW(τ) is simply W/S(t) which as t gets large will converge to the correct probability.

To understand how this works, draw a hypothetical D(t) plot, like that in the figure below. D(t) will always be below A(t) - a job obviously cannot experience a delay in the queue until after it has arrived. D(t) will cross S(t) horizontally when a service starts before τ and vertically when a job is delayed longer than τ.

Adding the Delay Event Count to Estimate the Waiting Distribution
Adding the Delay Event Count to Estimate the Waiting Distribution

See also Generalizing the Notion behind Little's Law (DELAY.MOD).



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