Uncertainty in the Demand for Service - Faculty of Industrial ...

Uncertainty in the Demand for Service:
The Case of Call Centers and Emergency Departments
M.Sc. Research Proposal
Shimrit Maman
Advisor: Prof. Avishai Mandelbaum
The Faculty Of Industrial Engineering and Management
Technion - Israel Institute of Technology
April 15, 2007
1

1 Introduction
During the last century and the beginning of the current one, the service sector has grown significantly
and now accounts for approximately 70% of the national income in the United States.
The service sector covers a wide spectrum of activities, e.g. education, professional services,
financial services and government services. In this thesis we focus on telephone call centers and
on the health care system.
Call centers are very commonly used by companies and organizations for managing their customer
relationships. This covers both the public and private sector. For some of the companies,
such as banks and cellular operators, their call centers are the main channel for maintaining contact
with their customers. In general, call centers are becoming a vital part of the service-driven
society nowadays. As a result call centers have also become an object for academic research.
For the analysis of call centers operations, queueing theory and statistics are being used. The
problems that call centers operators are dealing with, are often related to statistical characteristics
of the calls arrival and handling processes. Very often, about 70% of the operations costs
are devoted to human resources. As a result, forecasting the calls arrival rate, understanding the
handling process, setting the right service performance measures and staffing levels, are central
problems at all call centers. These are issues that any call center manager deals with on a daily
basis.
Another important area of the service sector is the health care system. Hospital managers
are increasingly aware of the need to use their resources as efficiently as possible in order to
continue to assure their institutions’ survival and prosperity. Moreover, there is a consistent
pressure, coming from the patients, to increase the quality of care by technique improvements
and medical innovations and to shorten the sojourn time in the hospital. Green [6] describes
the general background and issues involved in hospital capacity planning and provides examples
of how Operations Research models can be used to provide important insights into operational
strategies and practice.
Satisfactory customer service can be defined in many ways, based on various performance measures.
Focusing on operational measures, a customer enjoys satisfactory service if her delay in
queue is at most τ seconds [11]. For an emergency medical service system, the main performance
measure is the fraction of calls that are reached within some time standard. The response time
is typically considered to begin at the instant the call was made and end when an ambulance
reaches the call address. In North America, a typical target is to reach 90% of the most urgent
calls within 9 minutes [2]. The measures of quality are not absolute. They differ among service
system in accordance with the system’s goals, its environment and the services it offers.
The staffing problem is key for providing satisfactory service. Managers of call centers must
decide on how many agents to hire and get to work at any time in order to ensure satisfactory
customer service. In emergency medical service and in hospitals the system is even more complicated.
Staffing consists of scheduling decisions for ambulances and their crews, for doctors
and nurses and it must also account for the number of inpatient beds.

Service environments are typically very complex. One of the main complexity factor is the
uncertainty about the arrival rate: this uncertainty arises either because the parameter value
varies randomly over time or because it is unknown due to lack of information. Naturally,
staffing decisions must account for all this complexity, yet the challenge is to develop staffing
rules that are simple and insightful enough for implementation. For example, the ”square-root
staffing” rule is such a rule, as will be surveyed below.
Mathematical models have been developed to model the complexity of the call centers environment.
Their strength is their simplicity and the theoretical insights they provide. On the
other hand, their modeling scope is limited, and an analytical knowledge is needed in order to
apply them efficiently. These weaknesses could be the reason why these models are not being
used as often as expected. The most commonly-used models are the M/M/N queue (Erlang-C)
and M/M/N+M queue (Erlang-A), which will be described in section 2.
Computer simulation presents an alternative to mathematical models. When created and used
properly, simulation models can handle almost any model complexity and take into account
the very small details. There are limitations as well - the development and maintenance of
simulation models are expensive, and it can take hours to run a simulation even with today’s
computing capabilities.
A research trend in which mathematical models and simulation have been combined has recently
emerged. This in order to overcome the weaknesses and utilize the advantages of the two
methods. This is an approach adopted in this thesis.
The research proposal is structured as follows. In section 2 we describe some basic models
of service systems: (M/M/N and M/M/N+M ), and survey the work of [4]. In section 3 we
discuss issues related to uncertain arrival rates. A description of the research that we propose
to undertake is presented in section 4.
3

2 Some Basic Models with iid Customers and iid Servers
In this section, the two most common models that are used for call centers modeling and staffing
are presented. The first model is Erlang-C: first developed around 1910 by Erlang [3], it has
served until recently as the ”working-horse” of call center staffing. Its main deficiency is that
it ignores customers’ impatience, which is remedied by the second model, namely Erlang-A.
Impatience leads to the phenomenon of customers’ abandonment, and, already around 1940,
Palm [10] developed Erlang-A in order to capture it. We shall be using Erlang-A to motivate
three operating regimes for medium-to-large call centers: one which emphasizes service quality,
another that focuses on operational efficiency, and the third carefully balances these two goals
of quality and efficiency.
2.1 Erlang-C: The Square Root Safety Staffing Rule
The classical M/M/N (Erlang-C) queueing model is characterized by Poisson arrivals at rate
λ, iid exponential service times with an expected duration 1/µ, and N servers working independently
in parallel. One can view this model as a simple Birth-Death process where λ is
the (constant) birth rate and µ is the (constant) death rate. A Markovian-state description
of the process is the total number of customers in the system, either served or queued. The
corresponding transition rate diagram is then the following:
. . .
0 1 2 N-1 N N+1
2
N
N
Figure 1: Erlang- C as a Birth-Death Process
Erlang-C is ergodic if and only if its traffic intensity ρ = λ
µ < 1; ρ is then the servers’ utilization,
namely the long-run fraction of time that a server is busy.
The stationary/limit distribution is defined as
πj = lim
t→∞ P {L(t) = j}, j ≥ 0,
where L(t) is the system state at time t, namely, the total number of customers in the system.
Solution of the following steady-state equations yields the probabilities πj of being at any state
j during steady-state:
λπj = (j + 1)µj+1 0 ≤ j ≤ N − 1
(1)
λπj = (Nµ)πj+1 j ≥ N
The probability that in steady-state all the servers are busy is given by
πi, the stationary
probability of being in one of the states {N, N + 1, . . .}. This probability is sometimes referred
j≥N
N
. . .
4

to as the Erlang-C formula. It is denoted E2,N and is given by
E2,N =
j≥N
πN = (λ/µ)N
N−1 (λ/µ)
N!(1 − ρ)
i=0
i
−1 (λ/µ)N
+
i! N!(1 − ρ)
The Poisson distribution of arrivals has an important and useful consequence, known as PASTA
(Poisson Arrivals See Time Averages): it implies that the probability E2,N is in fact also the
probability that a customer is delayed in the queue (as opposed to being served immediately
upon arrival).
Using formula (2) and its relatives, one can calculate staffing levels N for any desired service
level, given the arrival rate and average service time. This can be easily done even with
a spreadsheet. However, such a solution does not provide any insight on the dependence of
N on model parameters, for example, how should N change if the load was doubled. Such
insight comes out of a staffing rule that goes back to as early as Erlang [3], where he derived
it via marginal analysis of the benefit of adding a server. (Erlang indicated that the rule had
been practiced actually since 1913.) This is the square-root safety- staffing rule, which we now
describe.
Let R = λ/µ denote the average offered load. Then the square root safety-staffing rule states
the following: for moderate to large values of R, the appropriate staffing level is of the form
5
(2)
N = R + β √ R (3)
where β is a positive constant that depends on the desired level of service; β will be refereed to
as the Quality-of-Service (QOS) parameter: the larger the value of β, the higher is the service
quality. The second term on the right side of (3) is the excess (safety) capacity, beyond the
nominal requirement R, which is needed in order to achieve an accepted service level under
stochastic uncertainty.
The form of (3) carries with it a very important insight. Let ∆ = β √ R denote the safety
staffing level (above the minimum R = λ/µ.) Then, if β is fixed, an n-fold increase in the
offered load R requires that the safety staffing ∆ increases by only √ n-fold, which constitutes
significant economies of scale.
What does (3) guarantee as far as QOS is concerned? For Erlang-C, this is the subject of
the seminal paper by Halfin and Whitt [7], where they provided the following answer:
Theorem 1: Consider a sequence of M/M/N queues, indexed by N=1,2. . . As the number of
servers N grows to infinity, the square-root safety-staffing rule applies asymptotically if and only
if the delay probability converges to a constant α (0 < α < 1), in which case the relation between
α and β is given by the Halfin-Whitt function
α = [1 + β/h(−β)] −1
here h(x) = φ(x)/(1 − Φ(x)) is the hazard rate of the standard normal distribution N(0,1).
(4)

Note that (3) applies if and only if √ N(1 − ρN) converges to β (β>0). Indeed, formally the
Theorem of Halfin-Whitt reads:
As N ↑ ∞, P (W ait > 0) = E2,N → α (0 < α < 1)
iff √ N(1 − ρN) → β (0 < β < ∞)
(equivalently N ≈ RN + β √ RN) .
In practice, this rule makes the life of a call-center manager easier: he or she can actually
specify the desired delay probability and achieve it by following the square-root safety staffing
rule (3), simply choosing the right β.
2.2 Erlang-A: Summary of Some Results
Trying to make the M/M/N model more realistic and useful, the following assumption is added:
each customer has limited patience, that is, as the waiting time in the queue grows the customer
may abandon. We assume that patience is distributed exponentially with mean 1/θ. This
model is referred to as Erlang-A (A for Abandonment). It is also a Birth-Death process, and its
transition rate diagram is depicted below.
. . .
0 1 2 N-1 N N+1
2
N
N
Figure 2: Erlang- A as a Birth-Death Process
The steady-state distribution is found by solving of the following equations:
λπj = (j + 1)µπj+1
0 ≤ j ≤ N − 1
λπj = (Nµ + (j + 1 − N)θ)πj+1 j ≥ N
The solution of the steady-state equations is given by:
⎧
⎪⎨
πj =
⎪⎩
(λ/µ) j
π0,
j!
j
λ (λ/µ) N
Nµ + (k − N)θ N!
0 ≤ j ≤ N
π0, j ≥ N + 1
where
π0 =
n
j=0
k=n+1
(λ/µ) j
j! +
∞
j
j=N+1 k=N+1
λ
(λ/µ) N
Nµ + (k − N)θ N!
N
. . .
2
−1
6
(5)

Just as with Erlang-C, appropriate staffing levels in Erlang-A can be found using the solution
to (4), but again, the disadvantage is that this does not provide any insights. To this end, the
Erlang -A analogue of Theorem 1 was proved in [5], and it is given as follows:
Theorem 2: Consider a sequence of M/M/N+M queues, indexed by N=1,2. . . . As the number
of servers N grows to infinity, the square-root safety-staffing rule (3) applies asymptotically if
and only if the delay probability converges to a constant α (0 < α < 1), in which case the relation
between α and β is given by the Garnett function
α =
1 + h(δ)/δ
h(−β)/β
−1
, −∞ < β < ∞. (6)
Moreover, the above conditions apply if and only if √ nP (Abandon) converges to some positive
constant γ that is given by
Formally, Theorem 2 reads:
h(δ)
γ = αβ − 1 , δ = β
δ µ/θ.
As N ↑ ∞, P (W ait > 0) → α (0 < α < 1)
iff √ N(1 − ρN) → β (−∞ < β < ∞)
(equivalently N ≈ RN + β √ RN)
iff √ NP (Abandon) → γ (0 < γ < ∞).
An important feature of Erlang-A is that, unlike Erlang-C, it is always stable whenever the
abandonment rate θ is positive.
Theorem 2 demonstrates that the square-root safety-staffing rule prevails for Erlang-A as well.
The QOS parameter β now depends on both the abandonment rate θ and the delay probability
α. It is significant that here β may take also negative values (since Erlang-A is always stable).
7

2.3 Erlang-A: Three Operational Regimes
Each organization has its own preferences in everyday functioning. Some of them try to get the
most from the available resources, while others see customers’ satisfaction as the most important
target. Depending on organizational preferences, three different operational regimes arise:
• Efficiency driven (ED).
• Quality driven (QD).
• Quality-Efficiency driven (QED).
As the number of servers increases, which is relevant for moderate to large call centers, these
regimes can be formally characterized by relating the number of servers to the offered load.
Efficiency Driven (ED) Regime: The efficiency driven regime is characterized by very high
servers utilization (∼ 100%) and relatively high abandonment rate (around 10%). In the ED
regime, the offered load R = λ/µ is noticeably larger than the number of agents N. This means
that the system would collapse unless abandonment take place. The formal characterization of
the ED regime is in terms of the following relationship between N and RED:
N = RED(1 − ε)
where ε > 0 is a QOS parameter: a larger value of ε implies longer waiting times and more
abandonment.
Quality Driven (QD) Regime: In the quality driven regime the emphasis is given to customers’
service quality. This regime can be characterized by relatively low servers utilizations
(for large call centers below 90%, and for smaller ones around 80% and perhaps less) and very
low abandonment rate. Formally, this regime is characterized by:
N = RQD(1 + ε), ε > 0
Quality and Efficiency Driven (QED) Regime: This regime is the most relevant for call
centers operation, since it combines a relatively high utilization of servers (around 95%) and low
abandonment rate (1%-3%). As noted already, the square-root safety-staffing rule (3) remains
valid in the model with abandonment, although now β depends not only on the service level
α = P {W > 0} but also on the abandonment rate θ. The number of servers in this regime is
given by the square-root formula
N = RQED + β RQED, −∞ < β < ∞ (7)
where β is, as usual, the QOS parameter; note that β can now take negative values, while
in the case of Erlang-C it is restricted to be positive in order to ensure stability.
8

2.4 Time-Varying Staffing to Achieve Time-Stable Performance [4]
In this section we extend our discussion to models which allow the arrival rates to vary with time.
The Mt/GI/∞ queueing model is characterized by Poisson arrivals at time-varying rate λ(t),
iid service times with cdf G, and infinitely-many iid servers working in parallel. The number of
busy servers at time t, L(t), has a Poisson distribution with a time-varying mean. The latter,
denoted R(t), can be also interpreted as the offered load at time t, and is expressed as follows:
t t
R(t) = E[L(t)] = E[λ(t − Se)] · E[S] = E λ(u)du = [1 − G(t − u)]λ(u)du, (8)
t−S
−∞
where
S is a generic service time;
Se is a random variable which describes the excess service time, with the following cdf:
P (Se ≤ t) = 1
E(S)
t
0
[1 − G(u)du], t ≥ 0.
One can also view the time varying offered load (8) of Mt/GI/∞ as the time-varying offered
load for the Mt/GI/Nt + GI model (with Nt iid servers working in parallel at time t and general
patience times), and in particular for the Mt/M/Nt + M model, namely the time-varying
Erlang-A model.
The authors of [4] show that to achieve time stable performance in the face of general arriving
rates it is enough to target a stable delay probability, which produces stable performance
of several other operational measures. More specifically, it is shown in [4] that for the timevarying
Erlang-A model (and in fact more generally), it is actually possible to reduce the hard
problem of time-varying staffing to an associated stationary staffing problem. Specifically, a
time-varying square-root staffing, i.e. N(t) = R(t) + β · R(t), applies (R(t) given in (8)) with
its characterizing parameter that is still the QOS parameter β (which, notably, does not vary
with time). This staffing level gives rise to a remarkable time-stable performance in which the
delay probability is constantly α. The relation between α and β in the time-varying model is
actually the Garnett function (6), derived for the stationary Erlang-A model.
Abandonment plays a crucial role in a call center functioning: one of the surprising consequences
of [4] is that staffing by the offered load itself is very effective in the QED regime. That
is, a special case of the square-root safety staffing rule (3), namely β = 0, actually reads
N(t) = R(t), t ≥ 0.
Of course, this would never work if no abandonment were allowed.
9

To show that the stationary Erlang-A model (M/M/N+N ) can indeed be used to predict the
performance of the time-varying Erlang-A model (Mt/M/Nt + N), the theoretical curve of the
expected waiting time conditioned on wait was compared to the simulation values of the waiting
time conditioned on wait for the time varying model. In the simulation, the mean value of the
time-varying arrival rate equals to 100 and the maximum deviation from the mean equals to
20. Figure 3 below, taken from [4], shows the comparison of these distributions. The empirical
results were obtained as an average of 5000 replications and show a remarkable fit to the theoretical
curve.
proportion
proportion
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
proportion
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
time
Wait|Wait>0 Garnett Function
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
time
Wait|Wait>0 Garnett Function
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time
Wait|Wait>0 Garnett Function
Figure 3: The empirical and theoretical conditional waiting time, given positive wait, for the
Mt/M/Nt + N model with three delay-probability targets: (1) α=0.1 (QD), (2) α=0.5 (QED),
(3) α=0.9 (ED).
10

3 Uncertain Arrival Rate
When designing service systems, we usually use stochastic models such as Erlang models, which
capture the uncertainty associated with the arrivals and service time. In addition, we assume
that these models’ parameters, such as the arrival rate and mean service time, are known. Then,
we analyze our system, computing performance measures and making staffing decisions under
these assumptions. However, very often the parameters’ values are not known with certainty.
This parameter uncertainty may lead to incorrect decisions. If we underestimate the true arrival
rate then we will not staff enough servers, which may be critical in an emergency medical service,
and if we overestimate the true arrival rate, we will provide better service than required, at the
cost of paying too many salaries.
Henderson [8] and Steckley et al. [11] consider two legitimate interpretations of the term ”uncertainty
of the arrival rate”. The interpretations differ in terms of whether the arrival rate
is randomly varying, or simply unknown. According to the first interpretation, on any given
day, the arrival process is modeled as a Poisson process with a random time-dependent rate
functions. The assumption of second interpretation is that the arrival rate function does not
vary from day to day, but it is not known with certainty. The main performance measure that
the authors of [8] the and [11] consider is the fraction of customers that wait in queue at most
τ seconds. They claim that the two systems can not be analyzed in the same way and show,
for each case, how to approximate this performance measure using steady-state approximations
and sketch how to estimate it using simulation .
For data to be generated by a Poisson distribution, the mean and the variance should be approximately
the same. However, in many practical cases, the data show a variance that significantly
higher than the mean. In order to account for this overdispersion, Koole and Jongbloed [9]
propose to model the arrival rate over [0,T ] by a Poisson mixture model in the form
λ(t) = λ0(t) · Z, 0 ≤ t ≤ T.
This model takes into consideration two sources of variability as follows. The scalar function
λ0(t) is such that T
0 λ0(u)du = 1. The value λ0(t) · ∆t is approximately the relative part of
expected arrivals during (t, t + ∆t) out of all arrivals, and it captures the predictable variability
in the arrival rate (time variations of expected arrival rate), e.g. peak demand of calls on
Mondays between 10:00 to 11:00, on a regular basis. In other words, the predictable variability
related to the shape of the arrival rate function. The second variable Z is a random variable that
captures stochastic variability: it describes the total number of arrivals to the service system,
i.e. describes the fluctuations around the time-dependent expected arrival rate. This model is
an example for the first interpretation of uncertainty, given by [8] and [11].
Every service manager must also take into consideration costs and how they are affected by
staffing decisions. Whitt [12] proposes methods for staffing a single-class call center with uncertain
arrival rate and uncertain staffing due to employee absenteeism while his goal is to maximize
the expected net return. He uses a deterministic fluid approximation and a numerical algorithm
for the conditional performance measures of M/M/N+N queue, and shows that for large arrival
rates the fluid model become very accurate and the exact numerical solution adds only little
11

practical value in terms of accuracy. Moreover, he shows that the quality of the fluid approximation
improves as variability of the arrival rate increases.
Uncertainty in future call volume complicates the process of determining staffing levels. It
is then important to correctly model the stochastic nature of call arrivals, and in particular, to
make corect predictions of call arrivals. Operation Researchers have been developing planning
models for service systems for the last few decades. Channouf et al. [2] refers to the emergency
medical system and survey the academic literature regarding the planning models topic.
Next, they develop and evaluate time-series models for estimating and forecasting daily and
hourly arrival rates, and then compare the models in term of their quality of fit and forecasting
performance using historical observations of a major Canadian city’s emergency medical service.
Implementing the QED’s ”square-root staffing” rule (7) requires that forecasted arrival and
service rates adhere to a certain level of precision. The requirements for prediction accuracy
is that the estimated arrival rate must not exceed an order of square-root deviation from the
real arrival rate. Otherwise the QED rule will not be effective since, with a larger error, it will
either grossly over- or under- estimate the staffing level. Aldor [1] develops two variations of
Poisson process models, which are based on a mixed Poisson process approach, for predicting
the system loads, and then develops different goodness-of-fit criteria that help determine the
model performance under the QED regime. Her results, of analyzing an Israeli cellular phone
company data, show that during busy periods, when the QED regime ”square-root staffing” rule
is relevant, the system will be able to perform at a desired level commensurate with the actual
load. The results gives some evidence that one can ensure a pre-specified QED regime using
load forecasts that are sufficiently precise.
12

4 Our Proposed Research
In order to demonstrate how uncertain arrival rate may influence staffing decisions, we first
consider an M/M/n+M queue with model-parameter uncertainty characterized by a random
arrival rate Λ. We let the mean service time and the mean patience time be 1, i.e. µ = θ = 1,
and we seek service level β which yields a time-stable delay probability α.
Assume Λ is a discrete random variable with finite or countably infinite set of values KΛ, and a
probability distribution pλi = P (Λ = λi). Let Ri = λi/µ denote the offered load in case Λ = λi,
and R the expected offered load
By the SLLN and the Garnett function (6):
α = lim
k→∞
where,
k
j=1 Wj
k
j=1 Aj
= lim
k→∞
1 k k j=1 Wj
1 k k j=1 Aj
R =
pλi · Ri .
KΛ
= EW
EA =
KΛ pi · Ri · αi
KΛ pi · Ri
=
KΛ pi · Ri · αi
R
Wj=Number of waiting customers at day j;
Aj=Number of arrivals at day j;
N−Ri
αi = P (W ait > 0|Λ = λi) = 1 − φ (since µ = θ; otherwise αi is the Garnett function
(6)).
√ Ri
Therefore, given α, we would like to find N. Then, assuming a square-root staffing rule, we
calculate the QOS parameter β = N−R √ .
R
Because of the analytical tractability of the Erlang-A model, using R, one can calculate the
value of β for each α according to (6). For analytically intractable models, one can use the ISA
simulation developed by [4] to obtain values of performance measures.
13
(8)

4.1 Examples
By comparing the model with and without the parameter uncertainty, we gain insight on the
effect of parameter uncertainty. The case of no parameter uncertainty arises when we replace the
random variable by its expected value. For example we let the arrival rate Λ have three possible
values, each with probability 1/3. We take the value 100 to be one of the possible arrival rates
as well as the expected offered load. Hence we have symmetric discrete distribution around 100.
Using R, we plot the probability of delay α as a function of the QOS parameter β (8) in Figure
4 below.
Alpha
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−10 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11
Beta
(10,100,190)
(25,100,175)
(40,100,160)
(55,100,145)
(70,100,130)
(85,100,115)
Garnett
Figure 4: Probability of Delay vs. Beta. The arrival rate gets one of 3 possible values with
equal probabilities, for example (85,100,115), (70,100,130) etc. as indicated above.
The first observation from Figure 4 is the growing deviation between the function of the random
parameter case and the Garnett function, as the variation of the arrival rate grows, when β is
positive. For negative values of β, the functions intersect and the connection between the variation
of the arrival rate and the function’s deviation from the Garnett function is unequivocal
in the context mentioned for positive β . Moreover, one should mentions that low staffing levels
will be good if Λ is low, but there will be a severe under-staffing if Λ turns out to be high. On the
other hand, high staffing level will be good if Λ is high, but there will be severe over-staffing if
Λ turns out to be low. A service system’s managers must take all these risks into consideration.
14

4.2 Future Work
• Updating the existing simulation software - our first step will be generalizing the
simulation code developed by Feldman [4] with the features of random arrival rate. We
will first check the results of the basic stationary Erlang-A model, and after that, we will
consider the expansion of the algorithm to the time varying Erlang-A model in the random
arrival rate case.
• Validate and understand theoretically Figure 4 - this step entails two things :
1. Establish the square-root staffing rule by comparing simulation results for random
arrival rate with theoretical results as in Figure 4, which are based on the square-root
staffing rule (3).
2. Understand theoretically the functions shape. Important and interesting things to
understand from the graphs are the intersection points of each function with the Y
axis (the case of β = 0, namely N = R), with the Garnett function and with the other
functions. In additional, we would like to understand the changes of the functions
from convexity to concavity, and the values of β for which α is stabilized.
• Arrival rate as a continuous random variable - we shall generalize our results to
arrival rates with densities. For example, we may let the prior distribution of Λ be Gamma
which is the conjugate distribution of the Poisson distribution. In other words, if we
sample Λ we get a posterior distribution which is another Gamma distribution with a
simple change of the parameters. Another continuous distribution which may be taken
into consideration is the Normal distribution, which is a natural choice when one has no
detailed analysis of the distribution of Λ. In doing so, we should assume that the standard
deviation of Λ is relatively small compared to its mean, so that negative values would occur
with only a negligible probability. However, the complexity of working with the normal
distribution must be taken into consideration.
• Theoretical results - motivated by the work of Halfin - Whitt [7] and Garnett [5], the
next step will involve developing a parallel theorem to those which are described in Section
2, in the case where the arrival rate is a random variable. For the M/G/N model (Poisson
arrivals, general service time distribution and N servers) there is, currently, no Halfin-
Whitt-like theorem. Hence, instead of mathematical analysis we shall use simulation, which
is applied to this model, in order to obtain similar results to the ones of Halfin-Whitt and
Garnett. Moreover, we shall consider approximations of performance measures that yield
insight as to their dependence on the model’s parameters. Part of these approximations are
diffusion approximations for stochastic processes, such as the process Q = {Q(t), t ≥ 0},
where Q(t) is the number of customer present in the system at time t. This kind of
approximation enables one to analyze the processes beyond steady-state.
• Considering other performance measures - we shall try to determine staffing levels
that will ensure time-stability of performance measures other than the delay probability,
for example the expected waiting time and the probability of abandonment.
• Optimization problems - we shall try to find staffing rules that, when applied, provide
minimum costs or maximum profit.
15

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