EAR Q.MOD

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1 Description
2 State Variables
3 Vertices
4 Initialization Conditions
5 Event Relationship Graph
6 English Translation
7 Comments

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=IATERL{1},STIME=MSTERL{1}
ENTER	Arrival of a customer	Q=Q+1, IATIME=ACORRIATIME+(RND>ACORR)(IAT*ERL{1})
START	Start of Service	S=0,Q=Q-1,STIME=SCORRSTIME+(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

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.