![](bg2.png)
dard queueing theory, utilising a Markov chain of
M/M/K queues. There have been countless other ap-
plications of these theories in problems as diverse as
highway traffic queues Evans et al. (1964) or manage-
ment of call centres (Aksin et al., 2007; Whitt, 2005).
There are some examples of queueing theory within
the problem of optimal staffing, such as Brooks et al.
(2011). They investigated the problem of care-at-
home services using a Markov decision process with
queues aiming to balance the cost of staffing against
the cost of rejection. In our model we assume that the
response service provider wishes to balance costs of
staffing against the response times, since the regulator
has incentivised that with targets. Another difference
between our approach and that of Brooks et al. (2011)
is that the inclusion of different classes of patients
resulted in the full problem becoming computation-
ally intractable. Consequently, they can only provide
heuristic bounds to guide the optimal policy. We aim
to keep our model relatively simple so that the solu-
tions can be found numerically in a reasonable time
frame.
So the unique contribution of this paper is to intro-
duce a time dynamic optimal control problem where
the control is the number of staff available, rather than
admittance or direction of calls on the system. This is
enabled by offsetting the expected total time between
receipt to arrival versus the cost of employing staff.
The result will be an optimal policy where the opti-
mal number of staff will be a function both of time
and the current state of the system. By including one
off costs at the start and the end of a shift, we can
ensure that the policy will result in realistic shift pat-
terns. The weighting of staff costs can then be varied
so as to establish the minimum staffing policy to meet
the required targets.
3 THE MODEL
Let us define a call to the service provider as a job
J which must be completed, and the staff member
that responds to that we shall denote as an engineer.
The engineer could be a policeman, fireman, ambu-
lance engineer, or a technician in the case of leakage
from a distribution network depending on the service
provider we are solving for. The different stages in
job completion are defined by the four times
• t
i
b
the time that the ith job is booked on the system.
• t
i
t
the time that the ith job is assigned to an engi-
neer.
• t
i
a
the time of arrival of the engineer at the ith job.
• t
i
d
the time that the engineer finishes the ith job.
Therefore, we define the ith job vector to be
J
i
= (t
i
b
,t
i
t
,t
i
a
,t
i
d
), (1)
and the set of jobs J = {J
1
,J
2
,..J
I
} represents all jobs
received over a fixed period of time. Furthermore, we
define η = t
a
−t
t
to be the travel time for an engineer
to reach their destination, and ξ = t
d
− t
a
to be the
total time which it takes for the engineer to finish the
job and become available again. We divide time into
K discrete periods of length ∆t, where the values of
K and ∆t may change depending on what we wish to
calculate. We can define the time at a given period to
be
t
k
= t
0
+ k∆t with k = 0, 1,..,K (2)
where t
0
is the reference time that is before the first
time/date in t
k
period of time, t
K
is after the last
time/date, and ∆t = (t
K
−t
0
)/K. Now define the num-
ber of jobs that have been received on the system
within the period t
k
to be B
k
, and the number received
but are yet to be completed at the time period t
k
as Q
k
.
Next we define the response time as the time between
the job being received on the system and the arrival
time of the engineer, or T = t
a
− t
b
. So to get an es-
timation of the response time, we must split the jobs
into three different types
• W the number of jobs waiting for the assignment
of an engineer.
• R the number of jobs where an engineer is in tran-
sit to reach a target
• A the number of jobs for which the engineer has
arrived but has not yet completed the job.
A job will count towards W if the call has been re-
ceived before (or at) the end of the current period, but
the engineer does not set off until after the end of that
period, counted in transit R if the engineer has set off
but not arrived, and counted as an arrived job A if the
engineer arrives before (or at) the end of the current
period and leaves after the start of the next period.
Obviously it follows that
Q
k
= W
k
+ R
k
+ A
k
, (3)
or Q is sum of all jobs waiting to be completed. We
impose no restriction on the number of jobs await-
ing completion but we must stipulate that the number
of engineers S available at time t
k
is greater than the
number in transit or at a job:
R
k
+ A
k
≤ S
k
. (4)
If we assume that the optimal way to manage the sys-
tem is to serve costumers on a first-come first served
basis, (although this might not always be the case if
S varies with time and we know some trips might
take a lot longer than others) then if R
k
+ A
k
< S
k
we
OptimalStaffingPolicy-AServiceSystemwithStochasticTravelTimes
111