IEEE Robotics & Automation Magazine - June 2015 - 58

Due to the complexity of considering patient pooling, the
case-mix optimization is only tested for the case without
patient pooling. In this case, d g becomes a scalar, N g (d g )
becomes a single-variable function, and the linearized model
can be simplified accordingly.
Performance Evaluation of a Patient Group
This section focuses on the performance evaluation of a given
group g with a set I g of patient types served by slot servers of
group g according to the given priority rule. The main purpose is to determine the minimal number N g of slot servers
needed and the actual BP r ig of each patient type with N g
slot servers.
We first present some simple properties of the merged
queue with pooled servers that help narrow the search space.
First, as no patient leaves before the completion of his/her
treatment sessions, the WTT requirements can only be guaranteed under the stability of the system, i.e., the relative server
workload is less than 100%, leading to
Ng 2

/ mi li .

i ! Ig

(15)

Another important property is the monotonicity of the BPs
with respect to the number N g of slot servers. As the priority rule is either FIFO or a static priority rule, it can be
shown that their treatment starting times are nonincreasing in
N g , leading to the monotonicity of the probability BPs r ig .
Thanks to the monotonicity, the determination of N g starts from
Each server is shared
some initial value and
then searches either
by different classes of
upward or downward
until it finds a minimal
patients with different
N g such that r ig # a i for
all patient types.
session durations.
The only remaining
issue is the evaluation of
the BPs for a given number N g of slot servers. Under the
ongoing assumptions, the system is similar to an M/D/n
queue if there is one single patient type and an M/G/n queue
if otherwise as the service time is the probability mixture of
the session numbers of different patient types. The major difference with the classical M/G/n queue is the discrete time
assumption of our system.
r ig

Discrete event simulation is used to evaluate the BPs as follows. At the beginning of each day, new patients arrive according to the Poisson rate m g with m g = R i ! I g m i . All patients
wait in a queue and are served according to the priority rule.
Consider first the case of an FIFO rule. When a patient is
selected to be served on a slot server, his/her type-i is randomly
generated according to probability distribution m i /m g , the service time is equal to l i, and the BP statistic of type-i is updated
according to the waiting time of the patient and ~ i . Consider
now a static priority rule. The type-i of each incoming patient is
randomly generated and he/she then joins the end of the
queues corresponding to his/her priority class. When a slot
server becomes available, it serves the first patient of the highest
priority queue that is not empty.
To reduce the variance and error of the simulation, common patient arrival streams of different patient types are
used for the evaluation of different patient groups with different slot server numbers. The simulation starts with a
warm-up period of 104 days followed by a simulation of T
days during which statistics are collected. Through some preliminary runs, we find T = 10 7 days to be a good balance
between precision and computational effort. It is used in all
numerical experiments.
To further speed up the computation, we use an M/M/n
approximation to determine the initial search point of N g .
The M/M/n approximation has a Poisson arrival rate m g
and an exponentially distributed service time with a mean
-1
equal to n -g 1 = ` / i m i n i j /m g . The BP of type-i patients
can be approximated by the following existing results of
M/M/n queues:
P ^W $ ~ ih = 1 - C ^n, ah e -^nn g -m g h~ i,
an
n! ^1 - th
C ^n, ah = n -1 k
,
n
a
/ k! + n!^1a - th
k =0
a/

< n,

(16)
(17)

(18)

where W is the steady-state waiting time of the M/M/n
queue, a = m g /n g is the so-called offered load, t = a/n is the
traffic intensity, and C ^n, ah is the so-called Erlang C formula
giving the waiting probability of an arrival patient. Equation
(18) ensures the stability of the system.
Table 1 compares the number of slot servers needed
obtained by the M/M/n approximation and the simulation
for the case without patient pooling. The M/M/n
approximation seems to slightly overestimate the
number of slot servers needed. It is used to set
Table 1. The number of slots in M/M/n approximation
versus simulation.
the starting N g for the case without patient
pooling. Without patient pooling, the starting
Maximum
N g is R ieI g n i, where, n i is the fewest slots of each
Arrival Service Breach
WTT
Rate
Rate
Probability (Days) Approximation Simulation
type. If the WTT requirements cannot be met
with this slot number, then the patient group is
2
0.04
5%
14
55
52
not improving and does not need to be consid0.2
0.1
5%
7
4
4
ered in the patient pooling optimization.

IEEE ROBOTICS & AUTOMATION MAGAZINE

JUNE 2015

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