288 9 Serial Limited Buffer Models
9.2.2 Structure of the State-Space
Each subsystem of the serial decomposition consists of an arrival generating ma-
chine (called the arrival-machine), a workstation processing machine (called the
service-machine), and a finite buffer of capacity w
max
−1 jobs in between the two
machines. A job in the service-machine counts as part of the work-in-process so
the subsystem has a capacity of w
max
jobs. The job being processed by the arrival-
machine does not count against the subsystem capacity limit because the job be-
ing served there is physically located in the previous workstation. The intent of
this section is the development of a queueing model of the steady-state occupancy
probabilities for the subsystem. Each service mechanism will be modeled as a GE
2
distribution.
Since a job is assumed to be always available at the arrival-machine, the machine
itself will either be processing a job in its first phase (remember, the machine is
considered to be a GE
2
system), processing a job in its second phase, or be finished
processing the job but the job is blocked because there is no room in the buffer.
For modeling purposes, it is necessary to keep track of the arrival-machine status
(i.e., either identify phase of processing or show the machine blocked), the service-
machine status (either identify phase of processing or show the machine idle), and
the number of jobs in the subsystem. Thus, a 3-tuple of information is needed to
represent the subsystem status. The continuous existence of a unit in the arrival-
machine does not match up with reality for the associated machine. The modeling
approach, however, is to account for the idle time for this real machine in the pro-
cessing time for the arrival-machine. Thus, this machine should be thought of as
the delay time between appearances of a job (inter-arrival time) to the workstation
under consideration. When the actual predecessor machine is idle, this time is part
of the inter-arrival time for the arrival-machine.
The 3-tuple state indicator is a vector with the first element representing the sta-
tus of the first node (arrival-machine), the second element defines the status of the
service-machine, and the third element is the total number of jobs in the subsystem.
As always, if at least one job is available for processing, the s ervice-machine will
be processing (not idle). Thus, the 3-tuple subsystem status vector is of the form
(a,s,w)
where the states for a are Phase 1, Phase 2 or completed processing but blocked
denoted by a ∈{1,2,b}. The states for s are similarly Phase 1, Phase 2, or idle
denoted by s ∈{0,1,2}, and the states for the third element of the three-tuple (work-
in-process) are w ∈{0,1,··· , w
max
}. Different subsystems are denoted by indexing
the 3-tuple elements by the subsystem index k as in (a
k
,s
k
,w
max,k
).
For each state where the machines are fully operational, there are 4 states asso-
ciated with each fixed work-in-process level. That is, since each machine can be in
one of two states, there are four combinations resulting: (1,1,w), (1, 2,w), (2,1,w),
(2,2,w), for 0 < w < w
max
. For the situation where the arrival-machine is blocked,
the buffer must be full and the service-machine must be busy; therefore, the possi-