MODEL:
! Model of a queue with arrivals in batches. In
this particular example, arrivals may show up in
batches of 1, 2, 3, or 4 units;
SETS:
! Look at enough states so that P( i) for large i
is effectively zero, where P( i) is the steady
state probability of i customers in the system;
STATE/ 1..41/: P;
! Potential batch sizes are 1, 2, 3 or 4 customers,
and A( i) = the probability that an arriving batch
contains i customers;
BSIZE/ 1..4/: A;
ENDSETS
DATA:
! Batch size distribution;
A = .1, .2, .3, .4;
! Number of batches arriving per day;
LMDA = 1.5;
! Number of servers;
S = 7;
! Number of customers a server can
process per day;
MU = 2;
ENDDATA
! LAST = number of STATES;
LAST = @SIZE( STATE);
! Balance equations for states where the number of
customers in the system is less than or equal to
the number of servers;
@FOR( STATE( N) N #LE# S:
P( N) * (( N  1)* MU + LMDA) =
P( N + 1) * MU * N +
LMDA * @SUM( BSIZE( I) I #LT# N: A( I)
* P( N  I))
);
! Balance equations for states where number in
system is greater than the number of servers, but
less than the limit;
@FOR( STATE( N) N #GT# S #AND# N #LT# LAST:
P( N) * ( S * MU + LMDA) =
P( N + 1) * MU * S +
LMDA * @SUM( BSIZE( I) I #LT# N: A( I) *
P( N  I))
);
! Probabilities must sum to 1;
@SUM( STATE: P) = 1;
END
