Consider a single workstation with
two classes of lots, one receiving high priority, the
other receiving low priority. Suppose that the workstation
contains only a single machine. The dispatch rule is
non-preemptive head-of-the-line, in that high priority
lots that arrive during the service of a low priority lot
do not interrupt service. Rather, the high priority lot
moves ahead of any low priority lots waiting for service.
Within a priority class, service is FIFO. Let lH,
lL
denote the arrival rates for the two classes. Let mH,
mL
denote the service rates for the two classes. Denote the
traffic intensities by rH =
lH/mH, rL =
lL/mL
and assume rH + rL
< 1. We obtain the limiting expected queuing delay
from Table 4.4 of Prabhu (1981), and after adding the expected service time, we find
the expected time in system for high priority lots to be
n (1 - rH)-1
+ mH-1,
and for the low priority lots to be
n ((1 - rH)(1
- rH - rL))-1
+ mL-1,
where
n = lL
mL-2
+ lH mH-2.
Suppose lH =
1 lot/hour, lL=
9 lots/hour,
mH = 12.0 lots/hour
= mL
Then n = 0.06944, and the expected time in system for high
priority lots is
[ .06944 / (1 - .0833)] + [1/12] = .0758 + .08333 =
.15909 hours = 9.545 min.
For low priority lots, the expected
time in system
is
[ .06944 / (1 - .0833)(1 - .0833 - .75)] + [1/12] =
.4545 + .0833 = .5379 hours = 32.27 min.
As we increase the percentage of high priority lots,
the expected time in system (or cycle time) of the low
priority lots increases rapidly, and non-linearly,
although the weighted average cycle time remains the same
(30 minutes in this example). This is shown in the figure
below.
