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Resource Optimisation for Cancer Pathways with
Aggregate Diagnostic Demand: a Perishable
Inventory Approach
Edilson F. Arruda∗1,2, Paul Harper1, Tracey England1,
Daniel Gartner1, Emma Aspland1, Fabr
´
ıcio O. Ourique3,
and Tom Crosby4
1School of Mathematics, Cardiff University, Senghennydd Rd,
Cardiff CF24 4AG, UK.
2Alberto Luiz Coimbra Institute- Graduate School and Research in
Engineering, Federal University of Rio de Janeiro, Rio de Janeiro
RJ, Brazil.
3Federal University of Santa Catarina, Ararangu´a SC, Brazil.
4Velindre Cancer Centre, Cardiff CF14 2TL, UK.
May 21, 2020
Abstract
This work proposes a novel framework for planning the capacity of
diagnostic tests in cancer pathways that considers the aggregate demand
produced by referrals from multiple cancer specialties (sites). The frame-
work includes an analytic tool that recursively assesses the overall daily
demand for each diagnostic test and considers general distributions for
both the incoming cancer referrals and the number of required specific
tests for any given patient. By disaggregating the problem with respect
to each diagnostic test, we are able to model the system as a perishable
inventory problem that can be solved by means of generalised GI/G/C
queuing models, where the capacity Cis allowed to vary and can be
seen as a random variable that is adjusted according to prescribed perfor-
mance measures. The approach aims to provide public health and cancer
services with recommendations to align capacity and demand for cancer
diagnostic tests effectively and efficiently. Our case study illustrates the
application of our methods on lung cancer referrals from UK’s National
Health Service. Healthcare Modelling, Capacity Planning, Inventory
Control, Queuing Systems
∗Corresponding Author: ArrudaEF@cardiff.ac.uk, efarruda@po.coppe.ufrj.br
1
1 INTRODUCTION 2
1 Introduction
The combination of an increased demand generated by the ageing of the pop-
ulation and frequent budget constraints leads to a constant need to optimise
healthcare resources (Capan et al., 2017). In particular, the demand for can-
cer care services in the United Kingdom (UK) has been constantly rising over
the last 20 years (Saville et al., 2019). That, in turn, gives rise to capacity is-
sues that have received increased attention from multiple government’s spheres
(e.g., Welsh Government, 2019; South Yorkshire, Bassetlaw & North Derbyshire
Cancer Alliance, 2019; Scottish Government, 2017; Meskarian et al., 2017).
The literature presents a number of alternatives for improving the manage-
ment of demand and capacity within healthcare services. For example, a proper
management of demand when a number of competing services are made avail-
able can be used to promote a better usage of the available capacity, while also
improving the quality of the service (e.g., Scottish Government, 2017; Arruda
et al., 2019). In contrast, scheduling and capacity allocation models can also
be implemented to promote an optimised use of the available capacity within a
prescribed planning horizon (e.g., Culpan et al., 2019; Woznitza et al., 2018).
Reviews in (Capan et al., 2017; Marynissen and Demeulemeester, 2019) provide
an overview of analytic and simulation tools for decision making in healthcare
systems in general. For more in-depth analysis of the literature on cancer care
management, see (Saville et al., 2019). In this paper we study a capacity plan-
ning problem for cancer diagnostic services considering the steady state demand
for diagnostic tests incoming from all available cancer specialties, hereafter re-
ferred to as cancer sites.
The complex nature of healthcare services gives rise to a complex supply
chain that can be viewed as a network of interacting services and supplies (Mar-
tins et al., 2019). Martins et al. (2019) discuss the importance of considering the
nature of these interactions in healthcare models, whilst also arguing that such
models often lack a “networks perspective”. In order to capture the interaction
of services and supplies, some authors opt to model the flow of services that are
part of the scope of their studies. This gives rise to complex analytic tools that
often resort to unrealistic simplifying assumptions, or to simulation tools that
are only able to compare a small number of alternatives (Saville et al., 2019;
Alagoz et al., 2011). The related literature includes a model of network com-
munity services in Canada (Bidhandi et al., 2019) that uses classical M/M/1
queuing models (e.g., Shortle et al., 2018) to estimate the probability of a delayed
service. A similar approach is applied by Wu et al. (2019) for multi-stage bed
allocation in hospitals. In contrast, Nguyen et al. (2018) propose a deterministic
approximation for a capacity allocation problem applied to an outpatient clinic.
Deterministic methods for capacity allocation in healthcare also appear in the
works of Nguyen et al. (2015) and Hulshof et al. (2013).
Related works also include material logistics applications in healthcare, which
were recently reviewed by Ahmadi et al. (2019). These applications include
inventory models with restrictive assumptions on the demand distribution (Ah-
madi et al., 2019; Rosales et al., 2015), which is often assumed to follow a Poisson
1 INTRODUCTION 3
process (Rosales et al., 2015). Another possible drawback of such models is that
they generally assume exogenous demand for supplies, an assumption that is vi-
olated in practice when the demand is indirect, such as the demand for surgical
supplies or diagnostic tests. While patient arrival is generally exogenous, the
demand for surgical supplies or diagnostic tests is endogenous and is generated
by patients that were previously admitted in the system. In a recent study,
(Richers et al., 2019) account for indirect demand in surgical supplies inventory.
This paper considers the problem of capacity planning for diagnostic tests
in the cancer pathway, considering the incoming demand for various cancer
sites. At the operational level, a patient typically requires multiple services and
appointments since being referred to a cancer pathway (Saur´e et al., 2012; As-
pland et al., 2019). While multi-appointment scheduling problems are reviewed
by Marynissen and Demeulemeester (2019), more specifically, Romero et al.
(2013) developed a simulation model for the treatment phase of skin cancer. A
spreadsheet simulation tool for a more general diagnostic and treatment unit
was introduced by Bowers et al. (2005), whereas Bikker et al. (2015) chose to
optimise the allocation of consultant doctor’s activities to accelerate the access
to radiotherapy treatments. Refer to Aspland et al. (2019) for a comprehensive
survey of clinical pathway modelling. Other applications of operational research
to cancer care are reported by Saville et al. (2019). They identified a gap in the
literature regarding optimisation methods for cancer diagnosis and staging. In
the present paper we seek to contribute to bridging this gap by addressing the
planning stage of the diagnostic and staging phases of a cancer pathway.
This study proposes an innovative analytic tool for capacity planning that
makes no assumption on the distribution of the cancer referrals, nor on the dis-
tribution of the diagnostic and staging tests required by a referred patient. In
order to keep this level of generality, we avoid a direct modelling of the path-
way, which has previously given rise to involved models that are very difficult
to solve (e.g., Saur´e et al., 2012; Castro and Petrovic, 2012). Instead, we exploit
the problem’s structure to simplify the model by disaggregating it with respect
to each diagnostic test. The rationale is somewhat similar to the demand dis-
aggregation in (Su´arez-Vega et al., 2017). It also bears some similarities to
agent-based approaches in which the agents examine the state of the system
and take decisions (e.g., Fuller et al., 2019).
By taking into account the aggregated demand for diagnostic tests generated
by each cancer site, we are able to produce a simple and easy to use analytic tool
that captures the essential characteristics of the problem whist also enabling the
user to optimise performance. For each diagnostic test, we start by assessing
the probability distribution of the demand produced by referrals of each cancer
site. We propose a recursive procedure to obtain such a distribution. Then, the
overall demand for the referred diagnostic test can be obtained as the convo-
lution of the demands of all cancer sites, which is also obtained by means of a
recursive procedure.
We consider the capacity planning problem for each diagnostic test as an
inventory problem with perishable inventory. At each day, the system possesses
a given capacity, which is viewed as an inventory of tests that can be used on
2 AGGREGATE DEMAND EVALUATION FOR A DIAGNOSTIC TEST 4
either incoming or queued patients. The inventory contains all the appoint-
ments that are available for a given test on a specific day. These appointments,
however, expire at the end of the day and will be no longer available in the
capacity inventory of the following day. Hence, unused capacity can be treated
analogously to expired inventory that needs to be discarded. Once we are in
possession of the probability distribution of the aggregated demand for tests,
the problem then becomes an inventory problem in which the decision maker
seeks a balance between unused capacity and the stationary distribution of the
number of queued tests. Once the capacity is defined, the problem can analo-
gously be seen as a GI/D/C queue that can be solved analytically, by means
of Z−transform (Chaudhry and Kim, 2003) or by means of embedded Markov
models (Shortle et al., 2018). Hence, the inventory model can be solved by de-
termining an adequate number of daily slots Cin the equivalent queuing model
that provides the desirable balance between unused capacity and waiting times.
The research and modelling approach is motivated by working with on-
cologists and managers across Wales as part of a wider Cancer Research UK
(CRUK) funded project. The Welsh Government has recently set out its ap-
proach to improving cancer services and outcomes with a significant focus on
the earlier detection of cancers and greater understanding and efficiency of the
pathways patients take. Subsequently, the Cabinet Secretary announced the
introduction of a single cancer pathway (SCP) and its implementation must be
properly tested and evaluated to understand the impact on patient care, treat-
ment outcomes and the wider health system. Our research is therefore aimed
at supporting Welsh Government in making recommendations to align capacity
to best match demand in an effective and efficient manner, and to ultimately
improve patient care and outcomes.
The remainder of this paper is organised as follows. Section 2 introduces the
problem and the recursive algorithm for the evaluation of the aggregate demand
per test. Section 2 introduces a recursive evaluation of the overall demand for
tests. Section 3 discusses capacity planning and the evaluation of the long-term
behaviour of the system for prescribed capacities. A case study to illustrate
the approach based on data from the U.K.’s National Health Service (NHS) is
presented in Section 4. Finally, Section 5 concludes the paper.
2 Aggregate Demand Evaluation for a Diagnos-
tic Test
Let us consider a set of n≥1 cancer sites that are being monitored and treated
in a given health service. To enter the pathway of a given cancer site, the patient
must be referred to the service by a physician. For each cancer site, the pathway
includes appointments with specialist consultants and medical teams, as well as
a set of specialised diagnostic and staging tests. The former are designed to test
for cancer, whereas the latter are tools to determine how advanced the cancer
is. In this paper, we are interested in determining the number of daily slots that
2 AGGREGATE DEMAND EVALUATION FOR A DIAGNOSTIC TEST 5
should be made available at any given day to satisfy the overall demand for each
diagnostic or staging test, in such a way as to ensure that the patient does not
have to wait excessively for an appointment should he or she be assigned the
test.
2.1 Aggregate Demand for a Single Cancer Site
Firstly, let us consider the demand for a given diagnostic test, produced by
incoming patients with cancer suspicion in a specific site. Each incoming patient
may or may not require this specific diagnostic test, but if they do require it,
they may take the test multiple times. Hence, the problem involves two random
variables, one that represents the number of incoming patients on a given day,
and another to denote the number of times that an incoming patient will have
to undertake the test under consideration.
Let Akbe a random variable representing the number of patient referrals
on any given day for cancer site k∈ {1, . . . , n}, which takes values from the
set ΩAk={0,1, ,...,N}. Let pAk(m), m ∈ΩAk, denote the probability that
exactly mpatients are referred for cancer site kon a given day. In addition,
define a random variable Tkto represent the overall demand for the test on a
given day that is produced by cancer site k. It is clear that the number of tests
performed is a function of the number of incoming referrals. Hence, by using
the total probability theorem, we have:
P(Tk=j) = X
m∈ΩAk
P(T=j|Ak=m)pAk(m).(2.1)
To simplify the notation, let Vmbe the total number of tests given that we
have exactly m∈ΩAkincoming referrals. Hence, P(Vm=j) = P(T=j|Ak=
m). Now, let the random variable Ydenote the number of tests required by
a single referred patient, which takes values from the set of integers ΩY=
{n1, , . . . , n2}, with n2≥n1, and denote by pY(i), i ∈ΩY, be the probability
that a patient will require exactly itests. From the definitions, it is clear that Vm
is the sum of mindependent and identically distributed (iid) random variables
Yl,1≤l≤m, with Yl∼Y, ∀l. Hence, Vmcan be seen as the convolution of
m iid variables, whose distribution can be obtained by the iterative procedure
below, by considering one convolution at a time, as follows:
P(Vm=i) =
i
X
j=(m−1)n1
P(Vm−1=j)pY(i−j), m ·n1≤i≤m·n2,∀m≥2,
(2.2)
with
P(V1=i) = pY(i),∀n1≤i≤n2.
Hence, Eq. (2.1) can be reformulated as
3 CAPACITY PLANNING 6
P(Tk=j) = X
m∈ΩAk
P(Vm=j)pAk(m),(2.3)
where the first term in the product is obtained from (2.2).
2.2 Considering Multiple Cancer Sites
Multiple cancer sites pose no significant additional difficulties for the calculation
of the distribution of the total demand for tests. In that case, one just needs
to repeat the procedure detailed in Section 2.1 for each cancer site. Like in
the previous section, let us assume that a total of n≥1 cancer sites make use
of the considered test, and recall that Tk, k = 1, . . . , n is the random variable
representing the total demand for tests from all incoming referrals for cancer
site 1 ≤k≤n. In that case, the total demand for tests is
T=
n
X
k=1
Tk.(2.4)
The distribution of Tis now the convolution of ndistinct and independent
probability distributions, and the iterative procedure to find the distribution of
Tis rather similar to that presented in the last section:
P(Wm=i) =
i
X
j=0
P(Wm−1=j)pTm(i−j), i ≥0,∀m≥2,(2.5)
with
P(W1=i) = pT1(i),∀i∈ΩT1,
and T=Wn.
3 Capacity Planning
This section discusses the elaboration of a capacity plan that makes use of the
aggregate demand for tests obtained by the procedures in Sections 2.1 and 2.2.
Let E(T) be the average daily demand for a given test. A simple course of
action for the decision maker is to offer a fixed number C∈Nof daily slots,
where Nis the set of natural numbers. Considering that a slot that is not
occupied on a given day cannot be kept in inventory for the next day, we can
model the resulting system as a GI/G/C queue. Furthermore, to ensure finite
waiting times and long-term stability, we must have C > E(T) (Shortle et al.,
2018). Such a system can be solved either analytically, using a Z−transform the
procedure detailed in Section 3.1, or by means of an embedded Markov chain
(e.g., Shortle et al., 2018), as described in Section 3.2. The choice of Cdepends
on the trade-off between unused capacity and quality of service to the end users,
which can be modelled in a number of ways; for example as a function of the
stationary distribution of the waiting times. Regardless of the long-term goal,
3 CAPACITY PLANNING 7
the decision maker has to be able to evaluate the steady state distribution of
the resulting queue as described in the following subsections.
Another possible approach, which will be explored in detail in the Case
Study - Section 4 - is to have some temporary extra capacity that is deployed
with a given probability. In that case, an analytic solution by means of the
Z−transform is no longer applicable. However, embedded Markov chains can
still be employed, as detailed in Section 3.2.
3.1 Steady State Distribution for the Overall Demand us-
ing the Z-Transform
The problem of finding steady state distribution of the resulting queuing system
for a given fixed capacity C > E(T) can be solved by means of the Z−transform,
making use of an equivalent signal processing formulation. Let pT(m) = P(T=
m) = tm, denote the probability that exactly mtests are requested on a given
day. The probability generating function (PGF) can be defined as
T(z) =
N
X
m=0
tmzm.(3.1)
The PGF T(z) is a polynomial of degree N, where Nis the maximal possible
number of test requests on a given day. The probability mass function (PMF)
can be expressed as a discrete sequence
t(m) =
N
X
r=0
trδ[m−r] (3.2)
that can be used to find the PGF
T(z) = Z{t(−m)},(3.3)
where Z{·} denotes the Z−transform operator (e.g., Oppenheim and Schafer,
2009). Let Xk, k ≥0 be the stochastic process that describes the number of test
requests in the system, i.e. the tests that are either being serviced or are waiting
to be processed at any given period k≥0. Then, the steady state probability
of process Xk, k ≥0, denoted by vector π, such that
πm=π(m) = lim
k→∞
P(Xk=m), m ≥0,
has PGF given by
Π(z) =
∞
X
m=0
πmzm=Z{π−m}.(3.4)
Bruneel and Wuyts (1994) and Chaudhry and Kim (2003) propose an explicit
expression for the generating function of the overall demand, given by
Π(z) = T(z)Q(z),(3.5)
3 CAPACITY PLANNING 8
where
Q(z) =
L
Y
i=1
1−βi
z−βi
,(3.6)
and βiis a root of zC−T(z) = 0. The polynomial zC−T(z) has L= (N−C)
roots outside of the unity circle |z| ≤ 1 (Bruneel and Wuyts, 1994; Chaudhry and
Kim, 2003). One can find Π(z) using the inverse Z−transform, and modelling
Eq. (3.5) as a digital causal stable filter whose output gives π−m. We have
Π(z−1) = T(z−1)H(z),(3.7)
where H(z) is the system function, and it has Lpoles pi=1
βi, i = 1, . . . , L,
within the unit circle, and
H(z) =
L
Y
i=1
1−pi
1−piz−1=ν0
1 + PL
i=1 ηiz−i.(3.8)
The steady state distribution for the requested tests in the system, π, is the
inverse Z−transform of Π(z−1). The digital filter output can be implemented
using the following difference equation (Oppenheim and Schafer, 2009):
πm=ν0tm−
L
X
i=1
ηiπm−i,(3.9)
where ν0is the numerator of Eq. (3.8), and ηiare the coefficients of the poly-
nomial in the denominator of Eq. (3.8).
3.2 Steady State Distribution for the Overall Demand via
Embedded Markov Chains
A G/M/C queue can be described by an embedded Markov chain (Shortle et al.,
2018) with state space S={0,1,2, . . .}that describes the evolution of process
Xk, k ≥0, described in the previous section as the number of test requests in
the system. Let pT(l) = P(T=l) be the probability that ltest requests are
received on a given day. The elements of the transition matrix PCfor a given
capacity Care given by:
pC
ij =
pT(l),if j=i−C+land i>C;
pT(l),if j=land i<C;
0,otherwise,
(3.10)
for all i, j ∈S. The first line in Eq. (3.10) models the transition when there is
already a queue of tests waiting to be processed. The second line considers the
system with no waiting test requests; observe that all requests already in the
system will be served in the current period, whilst the incoming requests will be
processed on the following day. Classical Markov chain theory yields that the
4 CASE STUDY 9
steady state distribution of Xk, k ≥0 is the solution of the following system of
equations (Br´emaud, 1999):
πP C=π,
∞
X
i=0
π(i) = 1.(3.11)
Now consider the case where the capacity is a random variable, i.e. Cis a
random variable taking values from the set of positive integers ΩC={c1, c1+
1, . . . , c2}, with P(C=c) = PC(c). In that case, the Z−transform method
of the previous section is no longer applicable. However, the Markov chain
approach is still valid, and the system is stable if E(T)< E(C). In that case,
we have
PC=X
l∈ΩC
pC(l)Pl,(3.12)
where Plis evaluated by means of Eq. (3.10), with C=l. Once again, the
steady state behaviour is obtained by solving (3.11).
The variable capacity setting is interesting to model the case when extra
capacity is made available with a given probability, for example, by granting
access to some shared resources. This possibility will be explored in the case
study in Section 4.
4 Case Study
We illustrate our novel methdological approach using data on lung cancer refer-
rals from one health board (region) of South Wales. We consider data on lung
cancer referrals from the Cwm Taf University Health Board in Wales, covering
the 26-week period from 1st April 2016 to 30th September 2016. A total of 341
patients were referred to the lung cancer pathway in the studied period.
In this example, we will consider CT test requests. Considering five business
days per week, we have a rate λ=341
26·5≈2.623 incoming patients per day. We
assume that the number of daily incoming referrals is described by a Poisson
process with rate λ. Imposing a boundary of 0.99 in the cumulative distribution
FA:A→[0,1], we limit the maximum number of arrivals to n= 7, and we
make
P(A= 7) =
∞
X
k=7
P(A=k),
to obtain the probability distribution depicted in Figure 1.
The total number of CTs performed over the period is 393. For the sake of
modelling, and considering the Optimal Lung Cancer Pathway guidelines (Lung
Clinical Expert Group, 2017), we assume that every patient undergoes at least
one CT test. Since there were 341 lung cancer referrals in the database, we
assume that 52 patients had a repeated CT scan. Hence, we have PY(1) =
341
393 ≈0.868, and PY(2) ≈0.132. Applying Eq. (2.3) to the available data, we
4 CASE STUDY 10
Incoming Patients
01234567
Probability
0
0.05
0.1
0.15
0.2
0.25 Arrival Distribution (Av. Rate = 2.6623)
Figure 1: Incoming Distribution of Lung Cancer Suspicions
obtain the distribution of the number of daily CT requests, depicted in Figure
2.
The cumulative distribution of CT requests per business day is depicted in
Figure 3 below. One can see that, for a service rate ρ≥0.8 a supply of five CT
slots per business day would suffice, whereas at least four CT slots per business
day are required for stability. Furthermore, depending on the required service
rate, the daily available capacity should range between the average number of
daily CT requests λCT ≈3.014 and 14 CT slots per day. A fractional capacity
can be attained by providing a fixed number of daily slots, for example 3 slots,
and offering some extra slots on specific days. For example, by having three
regular slots plus one extra slot every Monday, we would have an average of 3.2
slots per business day.
4.1 Optimal Capacity Planning
As mentioned in Section 3, if we keep a fixed capacity, that is, a fixed number
of daily CT slots, our model becomes a GI/D/C queuing system. In that case,
we may define Cas a function of a compromise between the perceived costs
of delayed tests and unused capacity. However, more general inventory policies
can be pursued, which would produce GI/D/C queuing systems with removable
servers. By convention, we assume that no incoming request can be processed at
4 CASE STUDY 11
CT Requests
0 2 4 6 8 10 12 14
Probability
0
0.05
0.1
0.15
0.2
0.25 CT Demand for Lung Cancer (Av. Rate = 3.014)
Figure 2: Distribution of CT requests for lung cancer referrals
the time of arrival. All requests have to wait at least one period to be processed.
The motivation is that a cancer test typically requires an appointment and very
seldom the appointment will be available for the same day.
Consider the example in the previous section, whose cumulative distribution
of requests is depicted in Figure 3, and whose average daily request rate is
λCT ≈3.014. Let us assume we have a fixed capacity of C= 3 daily CT
slots and an extra slot that can be used with a probability α∈[0,1]. For a
given fixed capacity C, the system can be described by a Markov chain (e.g.,
Br´emaud, 1999) with state space S={0,1, . . . }and transition matrix PC
defined in Section 3.2.
In the example, we have:
PC=
pT(0) pT(1) . . . pT(14) 0 0 0 0 . . .
pT(0) pT(1) . . . pT(14) 0 0 0 0 . . .
pT(0) pT(1) . . . pT(14) 0 0 0 0 . . .
0pT(0) . . . pT(13) pT(14) 0 0 0 . . .
0 0 . . . pT(12) pT(13) pT(14) 0 0 . . .
0 0 . . . pT(11) pT(12) pT(13) pT(14) 0 . . .
0 0 . . . pT(10) pT(11) pT(12) pT(13) pT(14) . . .
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
....
,
4 CASE STUDY 12
CT Requests
0 2 4 6 8 10 12 14
Cumulative Probability
0
0.2
0.4
0.6
0.8
1
1.2 Cummulative Distribution of CT Request
Figure 3: Cumulative Distribution of CT requests for lung cancer referrals
for C= 3. Observe in the transition matrix that, if there is no request which
cannot be immediately processed i.e. Xk=i≤C, then the next state equals
the demand in period k≥0, regardless of the current state i≤C. This happens
because the system will process all the ipending requests during period kand
will be left with only the incoming requests in the current period. In contrast,
when i > C, some of the requests will be left in the queue and the number of
impending requests at the onset of the following day will be Xk+1 =Xk−C+l,
where lis a realization of the random variable T, which represents the number
of incoming tests at any given time.
Now, let us assume in the example that we allocate an extra capacity with
a given probability α. Hence, we have P(C= 3) = 1 −αand P(C= 4) = α.
In that case, the transition matrix for process Xk, k ≥0, becomes:
PC=αP 4+ (1 −α)P3.
To evaluate the steady state behaviour of the system under any value of α,
it suffices to find the limiting distribution of Xk, k ≥0, by solving the system
in (3.11). Figure 4 depicts the steady state distribution for distinct values of α.
One can notice in Figure 4 that small values of αtend to keep a large queue of
impending tests. As we increase the value of αwe observe a steady decrease in
the system occupation, which is more steep for larger values of α.
Since the service is deterministic, we can easily obtain the expected waiting
RESOURCE OPTIMISATION FOR CANCER PATHWAYS 13
Queued Tests
0 10 20 30 40 50
Steady State Probability
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18 Steady State Distribution for Queued Tests
α ≈ 0.0142
α=0.05
α=0.25
α=0.50
α = 0.75
α=1
Figure 4: Number of pending tests
time of an incoming patient. Suppose, for example, that Xk=njust after the
arrival of the patient’s request. Then, the patient will have to wait dn
Cedays
for his request to be processed. Hence, we can easily calculate the steady state
distribution of the waiting time from the distribution of pending requests.
Figure 5 depicts the probability that an incoming request will have to wait
more than tdays to be processed for distinct integer values t≥0. With the
results in Figure 5 the decision maker can, for example, establish a target time
tand a target probability ¯p, and determine a suitable value of αsuch that
P(ω > t)≤¯p. For example, if we establish that at most 10% of patients should
wait more than 10 days for a test result, then α= 0.25 suffices. One the other
hand, α= 1 ensures that no more than 1 in a thousand requests will wait in
excess of six days. To facilitate the visualisation of the results, Figure 6 conveys
the same results in a semi-log scale.
RESOURCE OPTIMISATION FOR CANCER PATHWAYS 14
Wait time (Days)
0 5 10 15 20 25
P( ω > t)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Probabilities of Exceeding Wait Time Thresholds
α ≈ 0.0142
α=0.05
α=0.25
α=0.50
α = 0.75
α=1
Figure 5: Number of pending tests
RESOURCE OPTIMISATION FOR CANCER PATHWAYS 15
Wait time (Days)
0 5 10 15 20 25
P( ω > t)
10 -4
10 -3
10 -2
10 -1
10 0
Probabilities of Exceeding Wait Time Thresholds
α ≈ 0.0142
α=0.05
α=0.25
α=0.50
α = 0.75
α=1
Figure 6: Number of pending tests
5 CONCLUDING REMARKS 16
5 Concluding Remarks
Motivated by the need to improve the delivery of diagnostic services in cancer
pathways in Wales, this study introduced a novel recursive procedure to obtain
the probability distribution of the overall demand for a given diagnostic service
in a disaggregated manner. The approach is general in that it does not impose
constraints on the distributions of both incoming referrals and the number of
repeated diagnostic tests for each incoming patient.
By disaggregating the problem for each diagnostic test, we are able to model
the resulting system as a perishable inventory problem that can be solved by
means of a GI/D/C queuing model for a given capacity C. In that case, the
problem can be solved by means of analytic signal processing techniques or
Markov chain techniques, and the decision maker has to select a capacity C
that yields a good compromise between service quality and unused capacity.
However, when solved by means of Markov models, the approach is more general
and enables the decision maker to define random capacities by deploying shared
resources with a prescribed probability. The case study illustrates the flexibility
of the approach and demonstrates how the decision maker can use the results
to enforce bounds on the service time with prescribed probabilities.
Future research directions include introducing more flexibility to the deci-
sion maker by modelling the system as a Markov decision processes and thereby
defining the service level as a function of the current state of the system. While
flexible, such an approach requires the decision maker to prescribe a cost func-
tion, which may be very difficult in the setting, considering that the trade-off
between waiting times and the costs of extra capacity is not easily quantifi-
able. Moreover, since the actual system capacity is generally fixed, a flexible
use of such capacity by cancer pathways needs to be offset by an effective re-
source sharing protocol that enables the extra capacity to be deployed in other
pathways when it is not needed for cancer care. Hence, a number of research
problems can be defined to tackle each of these issues in future studies.
Funding
This work has resulted from research funded by a Cancer Research UK grant
‘Analysis and Modelling of a Single Cancer Pathway Diagnostics’ (Early Diag-
nosis Project Award A27882) and from a KESS2 grant under the project title
‘Smart Simulation and Modelling of Complex Cancer Systems’. Knowledge
Economy Skills Scholarships (KESS) is a pan-Wales higher level skills initia-
tive led by Bangor University on behalf of the HE sector in Wales. It is part
funded by the Welsh Government’s European Social Fund (ESF) convergence
programme for West Wales and the Valleys.
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