EAR Q.MOD

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Contents

Description

EAR_Q.MOD demonstrates the effects of dependent input by applying an exponential autoregressive (EAR) process (Lewis, 1981) to arrival and service times in CARWASH.MOD. The EAR process used has mean, M, and a correlation parameter of R. If you run EAR_Q.MOD with the same M but very different values of R, you will see a radical difference in the output series. Dependence in the service times has made the two systems behave very differently. When building this model, if we had looked only at the histograms of service times and ignored the serial dependence on service times, we might have had a very poor model.

State Variables

State Variables in EAR_Q.MOD
Variable Name Abbreviation Variable Description Size Type
QUEUE Q Number of customers in line 1 Integer
SERVERS S Server is available/busy = 1/0 1 Integer
STIME STIME Successive service times 1 Real
IATIME ATIME Successive interarrival times 1 Real
MST MST Mean service time 1 Real
IAT IAT Mean interarrival time 1 Real
SCORR SCORR Serial correlation between service times 1 Real
ACORR ACORR Serial correlation between successive arrivals 1 Real

Vertices

Vertices in EAR_Q.MOD
Vertex Name Vertex Description State Changes
RUN Initialization of the queue and the server S=1,IATIME=IAT*ERL{1},STIME=MST*ERL{1}
ENTER Arrival of a customer Q=Q+1, IATIME=ACORR*IATIME+(RND>ACORR)*(IAT*ERL{1})
START Start of Service S=0,Q=Q-1,STIME=SCORR*STIME+(RND>SCORR)*(MST*ERL{1})
LEAVE End of Service S=1

Initialization Conditions

Initialization Conditions in EAR_Q.MOD
Variable Description
QUEUE Initial number of customers in line
MST Mean service time
IAT Mean interarrival time
SCORR Serial correlation between service times
ACORR Serial correlation between successive arrivals

Event Relationship Graph

EAR_Q.MOD
EAR_Q.MOD

English Translation

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

The SIGMA Model, EAR_Q.MOD, is a discrete event simulation. 
It models A SINGLE-SERVER QUEUE (M/M/1) WITH EAR DEPENDENCE.
I. STATE VARIABLE DEFINITIONS.
For this simulation, the following state variables are defined:
QUEUE: NUMBER OF CUSTOMERS IN LINE   (integer valued)
SERVER: SERVER IS AVAIL/BUSY = 1/0   (integer valued)
STIME: SUCCESSIVE SERVICE TIMES  (real valued)
IATIME: SUCCESSIVE INTERARRIVAL TIMES  (real valued)
MST: MEAN SERVICE TIME  (real valued)
IAT: MEAN INTERARRIVAL TIME  (real valued)
SCORR: SERIAL CORRELATION BETWEEN SERVICE TIMES  (real valued)
ACORR: SERIAL CORRELATION BETWEEN SUCCESSIVE ARRIVALS  (real valued)
II. EVENT DEFINITIONS.
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(QUEUE,IAT,MST,SCORR,ACORR) event occurs when INITIALIZATION OF THE QUEUE AND SERVER.
   Initial values for, QUEUE,IAT,MST,SCORR,ACORR, are needed for each run.
   This event causes the following state change(s):
   SERVER=1
   IATIME=IAT*ERL{1}
   STIME=MST*ERL{1}
   After every occurrence of the RUN event:
   Unconditionally, INITIATE THE FIRST CUSTOMER ARRIVAL;
   that is, schedule the ENTER() event to occur without delay.
2. The ENTER() event occurs when ARRIVAL OF A CUSTOMER.
   This event causes the following state change(s):
   QUEUE=QUEUE+1
   IATIME=ACORR*IATIME+(RND>ACORR)*(IAT*ERL{1})
   After every occurrence of the ENTER event:
   Unconditionally, SCHEDULE THE NEXT ARRIVAL;
   that is, schedule the ENTER() event to occur in IATIME time units.
   (Time ties are broken by an execution priority of 6.)
   If SERVER==1, then START SERVICE WITH THE IDLE SERVER;
   that is, schedule the START() event to occur without delay.
3. The START() event occurs when START OF SERVICE.
   This event causes the following state change(s):
   SERVER=0
   QUEUE=QUEUE-1
   STIME=SCORR*STIME+(RND>SCORR)*(MST*ERL{1})
   After every occurrence of the START event:
   Unconditionally, THE CUSTOMER WILL GO INTO SERIVCE;
   that is, schedule the LEAVE() event to occur in STIME time units.
   (Time ties are broken by an execution priority of 6.)
4. The LEAVE() event occurs when END OF SERVICE.
   This event causes the following state change(s):
   SERVER=1
   After every occurrence of the LEAVE event:
   If QUEUE>0, then SERVICE THE WAITING CUSTOMER;
   that is, schedule the START() event to occur without delay.

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