Evaluating Tail Performance of Queueing SystemsUsing a Novel Stochastic Ordering Approach

Hung DoGrossman School of Business, The University of Vermont, Burlington, VT, hdo@uvm.edu

Masha ShunkoFoster School of Business, University of Washington, Seattle, WA, mshunko@uw.edu

Alan Scheller-WolfTepper School of Business, Carnegie Mellon University, Pittsburgh, PA, awolf@andrew.cmu.edu

In many settings the key metric for system performance is that system time does not exceed a crucial

threshold. Examples of such systems include an emergency response system where large risks are associated

with exceeding wait time thresholds, a network of storage/processing facilities that deal with perishable

items that expire if the system time exceeds a given threshold, and service centers in which calls/requests

are subject to service-level guarantees. Motivated by such systems, we propose novel stochastic orderings

that focus solely on the tail performance, namely, K-level stochastic ordering and K-convex ordering. With

these orderings, we extend the literature to establish that mean-preserved, threshold-structured flow control

policies provide Pareto improvement on multiple dimensions of interest, and more generally, the approach to

getting Pareto improvement is to reduce stochastically the queue while maintaining the expected throughput

rate. We then use our ordering to analyze queueing systems with multiple multi-server nodes under threshold-

structured flow control policies, and show that relative to our defined stochastic order system performance

under these policies is superior to performance under no control policies.

Key words : Stochastic orders, Tail performance, Queueing Systems, Flow Control Policies

1. Introduction

Using appropriate performance measures to make strategic and operational decisions is critical to

achieving good results. Selecting the correct measures depends on understanding the objectives

of decision-makers, which can vary greatly from case to case. We focus on situations in which

decision-makers are highly concerned with the performance in the tail of the distribution, i.e. in

avoiding extremely bad outcomes, rather than with the average performance overall. For example,

emergency medical professionals that are trying to save incoming patients may be concerned with

time exceeding the safe thresholds: it is essential for myocardial infarction patients to receive

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percutaneous coronary intervention within 120 minutes of arrival (OGara et al. 2013), incentivizing

healthcare providers to focus on reducing the probability that patients have to wait longer than 120

minutes rather than reducing the mean time to procedure overall. Likewise, project coordinators

who assign projects to different sub-contractors may face significant penalties if any project exceeds

its deadline; in such cases they would be interested in minimizing the probability of exceeding the

deadline, or minimizing the time beyond the deadline. Finally, processing operations that deal with

perishable items may face significant losses if items remain in system past a specific deadline.

Motivated by these situations, we develop new metrics for comparing the tail behavior of random

variables: K-level stochastic dominance and K-convex ordering. These metrics isolate the tail

performance, enabling us to evaluate and select policies that provide suitable performance in the

tail of the distribution. The current theory on stochastic orders allows one to compare distributions

stochastically over their entire domain (see e.g. Muller and Stoyan (2002)) and the theory on

bounded stochastic orders focuses on comparing conditional distributions (see e.g. Navarro et al.

(1997)), which essentially re-scale the distributions in the domain of interest. But, to the best of our

knowledge, there is no specific stochastic order that focuses on a subdomain without conditioning.

Conditioning implies dividing the mass in the tail by the probability of being in the tail, which may

favor policies that lead to unnecesssarily high probability of exceeding a threshold. Our proposed

order and corresponding metrics focus solely on the tail of the distribution without conditioning,

and hence allow decision makers to select policies that perform well in situations like the ones

described above. We illustrate the merit of this approach by evaluating the impact of flow control

policies with threshold structures on the performance of networks of multiple-server queueing nodes.

Do and Shunko (2015) propose this set of flow control policies with threshold structure, and show

that in a system consisting of multiple single-server queueing nodes, any policy within this set leads

to a stochastically smaller and less variable number of customers at each node, compared with the

benchmark of no flow control. This implies improvement with respect to performance measures

such as expected number of customers in the system, probability of observing a congested system,

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variance of the number of customers in the system, and expected number of customers in the system

given that the system is congested (or overcrowded). This result is not immediately extendable to

the system that has multiple multi-server nodes: In a single-server queueing system, for a given

expected arrival rate, application of Little’s Law indicates that the probability of no customer in

the system and probability of no wait stays the same regardless of the flow control policy. However,

in a queueing system with multiple-server nodes these probabilities depend on the flow control

policy, which significantly affects the analysis and, moreover, invalidates the result regarding the

stochastic ordering obtained in single-server case. Thus, the need for our novel stochastic orderings.

Specifically, in this paper we show that the number of customers at a multi-server queueing node

is not stochastically comparable to the number of customers under the benchmark policy of no

control with respect to the stochastic orders proposed in the literature. In contrast, we show that

the numbers of customers at a queueing node under the two policies are well-ordered with respect

to our newly defined orders. As a result, we provide a new managerial tool that lets decision-makers

identify policies that perform well in their domain of interest.

Our paper proceeds as following: in Section 2 we introduce a transformation technique that we

use in the subsequent analysis and present our stochastic orderings. In Section 3 we present the

model of our queueing system, describe the flow-control policy set, and demonstrate the difference

between the system with single-server nodes and multi-server nodes using numerical examples.

In Section 4 we characterize the stationary distributions in our model and present comparative

analysis of the numbers of customers in the system under the flow-control policies relative to the

benchmark of no flow control. We further demonstrate that our policies perform well according to

multiple practical performance measures. We summarize and conclude in Section 5.

2. Mathematical Preliminaries

For ease of exposition, we will present our results for non-negative real random variables, however,

our discussion in this section is extendable to real random variables.

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2.1. Definitions of stochastic orderings used

Let X be a continuous random variable with pdf f(x) and cdf F (x); and let K represent a critical

threshold, such that our domain of interest is x≥K. In this section we define stochastic orderings

that focus only on our domain of interest: [K,∞). Let X and Y be two random variables with

corresponding cdfs FX(x) and FY (y).

Definition 1. K-level first order stochastic dominance (denoted as ≤K−st):

X ≤K−st Y ⇔ FX(t)≤ FY (t) ∀ t≥K. (1)

This order implies that X is stochastically dominated by Y starting from K level (or in the right

tail), but ignores the order of X and Y at all levels less than K.

For analytical convenience we develop a transformation technique, head-uniform transformation,

that uniformizes the mass in the head of the distribution and hence lets us ignore the impact

of policies on the performance in the head of the distribution, isolating the tail characteristics.

Specifically, we define head-uniform transformation UK(X) such that the distribution in our domain

of interest (x≥K) remains untouched, while the mass in the head of the distribution (x<K) gets

spread evenly.

Let UK(X) be the random variable with the density function defined as follows, where FX(K−) =

limε→0+

FX(K − ε) (note that this definition allows us to include point K in the tail):

fUK(X)(t) =

FX (K−)

Kt <K;

fX(t) t≥K.

Notice that as opposed to the traditional conditioning approach, our transformation technique

does not rescale the tail of the distribution.

Next, we define several related operators that focus on the x ≥ K domain. Given a function

φ(x), define φK+(x) = Ix≥Kφ(x), where Ix≥K is an indicator function equal to 1 when x≥K, and

0 otherwise. Note that if φ(x) is increasing then φK+(x) is also increasing in the domain x≥K.

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Let EK+[φ(X)]def=∞∫K

φ(x)dFX(x) =E[φK+(X)]. Since the distribution in the domain x≥K does not

change after the transformation, we can easily show the following result: EK+[X] =EK+[UK(X)].

Next, in Lemma 1, we present two alternative equivalent representations of K-level first order

stochastic dominance that we will use in further derivations.

Lemma 1. Definition of K-level first order stochastic dominance is equivalent to:

X ≤K−st Y ⇔ EK+[φ(X)]≤EK+[φ(Y )] ∀ increasing functions φ(·); (2)

X ≤K−st Y ⇔ UK(X)≤st UK(Y ) . (3)

For convenience, we will use equivalence relation (3) most frequently in our presentation of the

following results and proofs. Next, we define increasing K-convex order and K-convex order using

the head-uniform transformation of X and Y .

Definition 2. Increasing K-convex order: X ≤K−icx Y if UK(X)≤icx UK(Y ).

Definition 3. K-convex order: X ≤K−cx Y if UK(X)≤cx UK(Y ).

2.2. Comparison with other stochastic orders

The K-level stochastic order is clearly weaker than the first order stochastic dominance; more

specifically, X ≤st Y ⇒ X ≤K−st Y . There are different stochastic orders related to bounded or

truncated random variables such as, e.g. failure rate order; however, they are different from our

approach in two important ways:

1. These definitions are based on conditional distributions, which rescale the tail by 1F (K)

while

our definitions leave the tail distribution untouched. This rescaling can distort the performance

measures and hence, obscure policies that perform well without rescaling.

2. These definitions may cover the whole domain, while our definitions only focus on the subdo-

main of interest (i.e., the tail of the distribution). Hence, our approach does not penalize policies

that do not perform well in the head of the distribution.

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3. System Model and Pareto-Improving Flow Control Policy Set

3.1. System Model

The system consists of n multi-server queueing nodes, where each arriving customer (or job) needs

to receive service at one of the nodes (e.g. a network of healthcare facilities serving one catchment

area, where each facility has multiple servers and can provide service to every patient; a network

of support centers, where each location has multiple operators that can provide customer service

for incoming calls; a network of storage facilities that can hold inventory in multiple subunits; a

network of parking facilities that can service one venue/event etc). Each node i has Mi servers

that operate with exponential service time characterized by rate µi. The arrival rate to node i

without flow control is Poisson with rate λ0i <µi, such that the total arrival rate to the network is

λ=∑i

λ0i . Let P be a set of stationary control policies; we will show in Section XXX that this set

is non-empty. Let P indicate a stationary control policy from the set P that can re-direct arrivals

from node to node based on the congestion (number of customers) at each node. Then, the arrival

rate to node i under control policy P is λi( ~NP), where ~NP is the vector of (NP0 ,N

P1 , ...,N

Pn ) that

represent the number of customers in the system (in queue and in service) at each node under

control policy P.

The expected arrival rate to node i under control policy P is:

λPi (k)def= E ~NP [λi( ~N

P)|NPi = k]. (4)

Since P is stationary by definition, λPi (k) is well defined. Notice that since the total arrival

rate to the network λ is fixed, increasing the expected arrival rate to one node will necessarily

decrease the expected arrival rate to at least one of the other nodes, which leads to negative

managerial implications, such as decreased expected revenue and/or decreased expected market

share. Similarly, decreasing the expected arrival rate to one node (e.g. a service agent) may lead

to an unfairly high load to another node raising unfairness concerns for the nodes in the network.

These are critical concerns for networks where nodes have individual decision making power - if the

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expected revenue and or market share is decreased, the manager of the node may decide to quit

the network. Hence, providing no change in the expected arrival rate to any node in the network

ensures sustainability of the network - neither node is discouraged from staying a member of the

network. Hence, to make sure that no node in the system is negatively impacted by the flow control

policy, we preserve the expected arrival rate as follows:

Condition 1 (Rate Conservation). E ~NP [λi( ~NP)] = λ0

i .

With this condition, the expected number of customers in the system can be represented in a

convenient way: Let πPi (k) be the stationary probability of having k customers at service node i

under policy P, then

Lemma 2. The expected number of customers in service at node i is:

∞∑k=0

πPi (k)min(k,Mi) =E[λi( ~N

P)]

µi= λ0

i

1

µi= ρ0

i . (5)

We will use this representation in our analysis. Notice the link between this result and Little’s

Law: isolating a set of servers as a system, the expected number of customers at the set of servers

is equal to the arrival rate to the servers times the expected time in service.

3.2. Policy objectives

Different systems and decision makers may use different performance measures to evaluate and

choose policies; hence, rather than picking a single objective function in our study, we analyze

policies based on the stochastic behavior of the tail of the distribution. Namely, we want the

policies to make the number of customers in the system stochastically smaller in the usual sense

and stochastically less in the convex order in the domain of interest (NPi > K). Policies that

possess such properties will perform well according to multiple practical performance measures:

For example, an objective function where cost increases convexly in the number of customers in the

system (the function is convex increasing in NPi ) and/or an objective where dis-utility increases

as the probability of number of customers exceeding allowed threshold increases (the function is

increasing in FNPi(K)).

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3.3. Flow control policies with threshold structure

We use a set of flow control policies with threshold structure introduced in Do and Shunko (2015).

Do and Shunko (2015) show that in queueing systems with multiple single-server nodes such policies

perform well according to multiple performance measures (expected number of customers, expected

number of customer in the congested state, probability of being overcrowded, variability of the

number of patients, and various composite measures) and compare the performance to different

practical benchmarks. In this paper, our primary goal is not to show the benefits of a threshold

policy; rather, we use the threshold policy set to demonstrate the merit of our proposed method-

ology - K-level stochastic orderings allow us to analyze this policy for a system with multi-server

queueing nodes. As a side benefit, we show that the benefits similar to the ones obtained in Do and

Shunko (2015) for single-server queueing system extend to multi-server systems. We compare the

threshold policy against the trivial policy of no flow control to demonstrate the marginal benefit

of any policy within this set according to our newly defined performance measurement approach.

Thus, we define the policy set PT representing coordinating mechanisms that satisfy Condition

1 and have the following threshold structure: There exists a threshold, Si, such that the expected

arrival rate to queueing node i is greater than the time-average arrival rate (λoi ) until this threshold

and lower than the time-average arrival rate above the threshold.

Definition 4 (Threshold policy set). For all P ∈PT , there exists a finite threshold Si > 0

such that: E[λi( ~N

P)|Ni = l]≥ λoi , ∀ l ≤ Si and ∃ l≤ Si s.t. E[λi( ~NP)|Ni = l]>λoi

E[λi( ~NP)|Ni = l]≤ λoi , otherwise,

and E[λi( ~NP)] = λ0

i (Condition 1 is satisfied).

The set PT represents a set of policies. In the next section, we will show a sample policy from this

set, which will demonstrate that the set PT is non-empty. For further comparison, we introduce

the benchmark policy without flow control - policy O. Notice that policy O 6∈PT because there is

no l such that E[λi( ~NO)|Ni = l]>λoi .

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There is a fundamental difference between a system with single-server queueing nodes and a

system with multi-server queueing nodes. In the case of a system of single-server queueing nodes

(Mi = 1 ∀ i), rate conservation (Condition 1) implies that the probability that a service provider is

idle under any flow control policy P is the same as under the no flow control policy O and equals

1−ρi, where ρi = λiµi

, since the expected arrival rate to a service provider is fixed; this follows from

Little’s Law. However, in the case of multiple-server queueing nodes (Mi > 1) this observation no

longer holds: the whole distribution (including the probability of observing zero customers) depends

on the selected policy and is hence, different from the distribution observed under O. We will later

show analytically the relationship between the probability of having no customers in system under

policies O and P ∈PT in Lemma 3.

Before presenting the technical analysis, to build intuition we will demonstrate numerically (using

a sample policy from Do and Shunko (2015)) the comparison of the distributions of the number of

customers in system under policies O and P ∈PT in section 3.4.

3.4. Numerical Illustration

We adapt a sample policy from Do and Shunko (2015) and use it to perform numerical experiments.

Here we briefly summarize the policy, an interested reader is referred to Do and Shunko (2015) for

details.

3.4.1. Sample Policy used for illustration Consider two service units with finite thresholds

S1 and S2 that indicate some capacity measurement (Si > Mi, for i ∈ {1,2}, where Mi is the

number of servers at node i). The arrival rate to node i under the benchmark (no re-routing, no

coordination) scenario is λoi for i ∈ {1,2}. Let li represent the number of customers at unit i for

i∈ {1,2}, and dij represent the re-routing rate from unit i to j for i and j in {1,2}. The re-routing

rates d12 ∈ [0, λo1] and d21 ∈ [0, λo2] are used as outlined in Table 1. We denote a sample policy from

this set with P(d12,d21)D – each policy is Markov and stationary, P(0,0)

D is equivalent to O. We let

PCD represent all policies that follow routing rules specified in Table 1 and that have (d12, d21)

such that rate conservation Condition 1 is satisfied.

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Node 2

Node 1

l2 <S2 l2 ≥ S2

l1 <S1

Rate to node 1: λo1 Rate to node 1: λo1 + d21

Rate to node 2: λo2 Rate to node 2: λo2− d21

l1 ≥ S1

Rate to node 1: λo1− d12 Rate to node 1: λo1

Rate to node 2: λo2 + d12 Rate to node 2: λo2

Table 1 Sample flow control policy P(d12,d21)D .

Lemma 1 of Do and Shunko (2015) shows that there exist d12 ∈ (0, λc1] and d21 ∈ (0, λc

2], such

that P(d12,d21)D ∈PCD and PCD ⊂PT for the system with single-server queueing nodes, their proof

can be easily extended to the system with multi-server queueing nodes considered here, which

demonstrates that set PT is non-empty.

3.4.2. Numerical Example Consider the following example: Two service units (1 and 2)

with the following parameters: λ01 = 9, λ0

2 = 13.5, µ1 = 1, µ2 = 1, M1 = 10, M2 = 15, S1 = 10, S2 = 15,

and d21 = 6.75. We find d12 = 2.53, s.t. P(d12,d21)D ∈PCD. We plot the probability and cumulative

distribution functions in Figures 1 and 2 respectively under two policies: sample policy P(2.53,6.75)D

and benchmark policy O.

Notice several interesting observations regarding the ordering of the distributions under the

threshold and benchmark policies. First, the probability of having no customers in the system is

higher under the benchmark policy than under a threshold policy (see zoomed area in Figures 1

and 2) - this is fundamentally different from the system with single-server queueing nodes where

these probabilities are equal. Second, the number of customers under the threshold policy is not

stochastically smaller over the full domain: The cdf of the number of customers under the threshold

policy P(2.53,6.75)D crosses the cdf of the number of customers under the benchmark policy from

below. We also note that the crossing point is below the threshold level S1 = 10 and hence, the

similar graphs of the U-transformed number of customers in the queue (Figures 3 and 4), where

the mass to the left of S1 is spread uniformly across all values in the head of the distribution,

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0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Number of customers in the system

Prob

abilit

y D

istri

butio

n Fu

nctio

n

Threshold PolicyBenchmark Policy

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

Benchmark Policy

Figure 1 The pdf of the number of customers under the threshold policy P(2.53,6.75)D crosses the pdf of the number

of customers under the benchmark policy twice.

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of customers in the system

Cum

ulat

ive

Dis

tribu

tion

Func

tion

Threshold PolicyBenchmark Policy

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2Benchmark Policy

Figure 2 The cdf of the number of customers under the threshold policy P(2.53,6.75)D crosses the cdf of the number

of customers under the benchmark policy from below.

illustrate that the U-transformed random variables are stochastically ordered, while the original

(untransformed) distributions are not.

In the next section, we formalize these observations and prove them analytically.

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0 5 10 15 200.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Number of customers in the system

Tra

nsfo

rmed

Pro

babi

lity

Dis

trib

utio

n F

unct

ion

Threshold PolicyBenchmark Policy

Figure 3 The pdf of the U-transformed number of customers under the threshold policy P(2.53,6.75)D crosses the

pdf of the U-transformed number of customers under the benchmark policy once.

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of customers in the system

Tra

nsfo

rmed

Cum

ulat

ive

Dis

trib

utio

n F

unct

ion

Threshold PolicyBenchmark Policy

Figure 4 The cdf of the U-transformed number of customers under the threshold policy P(2.53,6.75)D is above the

cdf of the U-transformed number of customers under the benchmark policy indicating that the latter

(U10(NO)) stochastically dominates the former (U10(NP(2.53,6.75)D )).

4. Policy Analysis

In this section we analyze policies from set PT and the benchmark policy O (Section 4.1), and

then, stochastically compare them in Section 4.2. We show that the number of customers under a

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flow control policy in set PT and that under the benchmark policy O are not comparable under

the stochastic order in usual sense, nor under the convex order (Muller and Stoyan 2002). However,

they are well-ordered under K-level stochastic order and under K-convex order as defined in Section

2. We then compare the policies with threshold structure with the benchmark with respect to

different performance measures in Section 4.3 to demonstrate the benefit of our approach: our

measurement perspective allows us to identify policies that perform well according to multiple

practical performance measures, while this is not apparent using the existing stochastic orderings

commonly applied in literature.

4.1. Stationary distributions under policies from set PT and under the benchmark policy O

For policy P in set PT , using balance equations, we can express the stationary probabilities as

follows:

πPi (m) = πPi (0)

m−1∏k=0

λPi (k)

m!µmi∀ 1≤m≤Mi; (6)

πPi (Mi +m) = πPi (Mi)

m−1∏k=0

λPi (Mi + k)

(Miµi)m= πPi (0)

Mi−1∏k=0

λPi (k)

Mi!µMii

m−1∏k=0

λPi (Mi + k)

(Miµi)m∀ m ≥ 1. (7)

The normalization condition implies:∞∑k=0

πPi (k) = 1.

And hence, summing up equations 6 and 7, we get:

πPi (0)

1 +

Mi∑m=1

m−1∏k=0

λPi (k)

m!µmi+∞∑m=1

Mi−1∏k=0

λPi (k)

Mi!µMii

m−1∏k=0

λPi (Mi + k)

(Miµi)m

= 1.

Rate conservation (Condition 1) then implies:

E[λi( ~NP)] =

∞∑k=0

πPi (k)λPi (k) = λ0i ⇒

πPi (0)

λPi (0) +

Mi∑m=1

m−1∏k=0

λPi (k)

m!µmiλPi (m) +

∞∑m=1

Mi−1∏k=0

λPi (k)

Mi!µMii

m−1∏k=0

λPi (Mi + k)

(Miµi)mλPi (Mi +m)

= λ0i .

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The benchmark policy with no flow control is denoted with superscript O. Under this policy, the

stationary probabilities (Equations 6 and 7) can be simplified as follows:

πOi (m) = πOi (0)(λ0

i )m

m!µmi∀ 1≤m≤Mi;

πOi (Mi +m) = πOi (Mi)(λ0

i )m

(Miµi)m= πOi (0)

(λ0i )Mi

Mi!µMii

(λ0i )m

(Miµi)m∀ m ≥ 1.

Again, the normalization constraint then implies:

πOi (0)

(1 +

Mi∑m=1

(λ0i )m

m!µmi+∞∑m=1

(λ0i )Mi

Mi!µMii

(λ0i )m

(Miµi)m

)= 1. (8)

And rate conservation (Condition 1) is equivalent to:

πOi (0)

(λ0i +

Mi∑m=1

(λ0i )m+1

m!µmi+∞∑m=1

(λ0i )Mi

Mi!µMii

(λ0i )m+1

(Miµi)m

)= λ0

i . (9)

4.2. Stochastic comparison of policies

In the next two lemmas, we compare policy O to policies in set PT at two points of interest: (1)

no customers in system and (2) number of customers in system equal to the threshold Si, which

later allows us to order the number of customers under these policies stochastically. First, we prove

our observation from Figure 1 that under any flow control policy P from set PT , the probability

of observing no customers in the system is lower than under the benchmark policy O.

Lemma 3. For any control policy P ∈PT with Si ≥Mi, the probability of having no customers at

queueing node i under policy P is less than that under policy O:

πPi (0)<πOi (0) ∀ P ∈PT . (10)

This result is intuitive since under a threshold policy, the servers are used more efficiently - they

are busier in less-crowded states and less busy in more-crowded states. Hence, the probability of

having an empty system is lower under the threshold policy. Next, we look at the probability of

having Si customers at unit i and show that it is lower under policy P from set PT than under

the benchmark policy O for all Si ≥Mi.

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Lemma 4. For any control policy P ∈PT with Si ≥Mi, the probability of having Si customers at

queueing node i under policy P is greater than that under policy O:

πPi (Si)>πOi (Si) ∀ P ∈PT . (11)

Using Lemmas 3, and 4, we can now state our main result:

Theorem 1. For any control policy P ∈PT with Si ≥Mi,

1. The cumulative probability of having no more than Si customers at queueing node i under

policy P is greater than that under policy O:

FNPi(Si)>FNOi

(Si) ∀ P ∈PT ; (12)

2. The number of customers in the system under policy P is smaller than under policy O in

Si-level stochastic order for any Si greater than or equal to capacity Mi:

NPi <Si−stNOi ∀ Si ≥Mi and ∀ P ∈PT . (13)

Theorem 1 indicates that any control policy in set PT decreases the cumulative probability of

observing high number of customers in the system (beyond threshold Si) and establishes that the

numbers of customers in the system under policy P and under policy O are well ordered according

to the Si-level stochastic order. This stochastic ordering further implies that any control policy

in set PT will improve the performance of the system according to any objective function that is

monotone increasing in the number of customers in the system in the domain of interest (NPi ≥ Si),

e.g. cost associated with being in overcrowded state, as discussed above in Sections 3.2.

When Si =Mi, the right tail of the distribution represents the queue size. Let QPi be the queue

size of the queueing node i under the coordination policy P. Using =st to indicate equality in

probability, we note that:

QPi =st [NPi −Mi]+. (14)

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Note that the stationary distribution of the queue size QPi , denoted with πPi (l), could be derived

from the distribution of the number of customers (NPi ) as follows:

πPi (l) =

Mi∑k=1

πPi (k) , l= 0;

πPi (l+Mi) , otherwise.

Using Theorem 1 and Definition 1, we then infer:

πPi (0)> πOi (0),

and

FQOi(k)<FQPi

(k) ∀ k.

Hence, we obtain the following result:

Corollary 1. QPi <st QOi ∀ P ∈PT .

As implied by Corollary 1, in addition to comparing the number of customers in system, our

approach allows one to also stochastically order queue size distributions under different policies.

So far, we have demonstrated that the threshold policy performs well according to the K-level

stochastic order, which implies that the policy will perform well according to any objective function

that is monotone increasing in the number of customers in the system. Next, we will show that

the threshold policy also performs well for any convex function of the number of customers in the

system.

Due to the queueing setting, we work with the discrete random variable (number of customers

in the system). In the literature, the convex order is defined over convex functions (see e.g. Muller

and Stoyan (2002)), while convex functions, by definition, are continuous and differentiable almost

everywhere (i.e., except on a zero measure set). Hence, convex order is not directly applicable to

discrete random variables. In this light, we utilize the following approach: we use the transformation

technique T introduced in Do (2012) to transform the discrete distribution of USi(NPi ) into its

continuous analog T [USi(NPi )] and work with this transformed variable.

17

Definition 5. (Do 2012) Given a discrete (nonnegative integer) random variable Y with mass

function pY (z) where z ∈ Z+ (nonnegative integers), T (Y ) is a nonnegative continuous random

variable with the right-continuous density function defined as pT (Y )(t) =∑z∈Z+

pY (z)Iz≤t<z+1, where

Iz≤t<z+1 is an indicator function equal to 1 when z ≤ t < z+ 1, and 0 otherwise.

The transformation operator T has several desired properties that we will use in further analysis:

Property 1. (Shown in Do (2012)) E[T (Y )] =E[Y ] + 12;

Property 2. T (UK(Y )) =st UK(T (Y )).

Define the difference between the expected U-transformed number of customers under the bench-

mark policy O and the expected U-transformed number of customers under any policy from thresh-

old policy set PT as:

∆Pi ,E[USi(NOi )]−E[USi(N

Pi )]. (15)

We can now state the following:

Theorem 2. 1. The difference between the expected U-transformed number of customers in the

system under policy P ∈PT and U-transformed number of customers in the system under policy

O is positive: ∆Pi > 0 ∀ i;

2. The T -transformed number of customers in system PT , shifted by ∆Pi , is smaller in Si−convex

order than the T -transformed number of customers in system under benchmark policy O:

T (NPi ) + ∆Pi <Si−cx T (NOi ) ∀ P ∈PT , ∀ i, and ∀ Si ≥Mi.

This result indicates that the T -transformed number of customers shifted by ∆Pi under policy

P ∈ PT is less variable and hence performs better for any convex cost function in the domain

[Si,∞). In the next section, we will show how our stochastic ordering results imply improvement

according to several practical performance measures defined on the original untransformed random

variables.

18

4.3. Performance measures

In this section we focus on several specific measures that can be used to assess performance of flow

control policies. All these measures possess the functional properties listed in Section 3.2 and are

motivated by practical applications.

1. Probability that the number of customers at a queueing node i under policy P is greater than

threshold Si:

FNPi(Si). (16)

For example, in managing emergency departments, this metric can capture the probability of

overcrowding in the department.

2. Any disutility function g(NPi ) that is monotone increasing in NPi in the domain of interest

NPi ≥ Si. As a special case, we discuss the average number of customers when the service provider

is over threshold Si:

ESi+[NPi ] =

∞∫Si

xdFNPi(x). (17)

If Si is equal to the number of servers at queueing node i, this measure represents the average

number of people in queue. If Si represents the limit that the system can handle without penalty

(e.g. by space limitation, or by licensing agreement), ESi+[NPi ] represents the number of customers

that will cause extra cost or risk. In inventory management, ESi+[X]−SiFX(Si) represents expected

backorder, where X represents the demand and Si is equivalent to the order quantity.

3. Conditional expectation of the number of customers when the service provider is over a

threshold that is defined relatively to a given service level α at queueing node i. Such measure

(Conditional-Value-at-Risk) as defined in Rockafellar and Uryasev (2002) is commonly used in

financial risk analysis. In particular, we use the CV aR− measure defined in Rockafellar and Uryasev

(2002) that can be applied to discrete random variables: in our analysis, the random variable of

interest (number of customers at a queueing node LPik ) is discrete.

Let α∈ (0,1) represent a service level such that the queueing node k is interested in not exceeding

the corresponding number of customers captured by Value-at-Risk, Si. For a continuous random

19

variable, Value-at-Risk Sαi [X] can be defined as F−1X (α). However, for a discrete random variable,

α may “split the atom”, such that FX(x) < α < FX(x + 1) indicating the need for a modified

definition of Sαi [X]: Sαi [X] = min{x : FX(x)≥ α} is the left-continuous inverse of FX .

Then, for a given value of α, we use a risk measure that captures the expected number of

customers at a queueing node conditional on exceeding the threshold Sαi [X]:

CV aRα[NPi ] =E[NPi |NPi ≥ Sαi [NPi ]]∀ i.

CV aR is traditionally defined as CV aRα[X] = E[X |X ≥ F−1X (α)] (Pflug 2000). Notice that for

a continuous random variable, CV aRα[X] = CV aR−α [X], but for a discrete random variable, due

to potential “atom splitting” – it is not. Hence, since our analysis includes both discrete and

continuous random variables, we use the CV aR−α [X] definition for consistency.

Remark 1. We would like to highlight the difference between the last two measures (ESi+[NPi ]

and CV aRα[NPi ]) that is important in evaluating policies. CV aRα[NPi ] measures the expected

value of NPi conditional on NPi >Si. This implies that if CV aRα[NPi ] decreases, it may be caused

by two drivers: expectation in the tail decreases and/or probability of being in the tail increases.

Hence, a low CV aRα[NPi ] value for a policy P may imply that the probability of exceeding the

threshold is high under this policy, which may be not desirable in business settings. In such cases,

ESi+[NPi ] and FNPi(Si) may be more useful for the decision maker. For completeness, we include

all three measures in our further analysis.

To show improvement in CV aRα[NPi ], we need the following intermediate lemma that follows

immediately from the head-uniform transformation definition:

Lemma 5. CV aRα[NPi ] =CV aRα[UK [NPi ]] ∀ α≥ FNPi (K).

We can now show that any policy P from set PT performs well according to all three measures

defined above.

Proposition 1. Policy P ∈PT outperforms policy O according to

20

a. the probability that the number of customers at service node i exceeds threshold Si >Mi

FNPi(Si)< FNOi

(Si);

b. any objective function g(X) that is monotone increasing in X in the domain X >Si and, as

a special case, according to the expected value in the tail:

g(NPi )< g(NOi ) and ES+i [NPi ]<ES

+i [NOi ];

c. the conditional expectation in overcrowded state: CV aRα[NPi ] + ∆Pi < CV aRα[NOi ] ∀ α ≥

FNPi(Mi) for each queueing node i.

Parts a. and b. follow directly from Theorem 1. In order to show part c., note that Theorem 2

implies CV aRα[T [U [NPi ]]]+∆Pi <CV aRα[T [U [NOi ]]]. Using Lemma 2 from Do and Shunko (2015)

that shows that CV aRα[T [X]] = CV aRα[X] + 12

for any X, this implies that CV aRα[U [NPi ]] +

∆Pi <CV aRα[U [NOi ]]. We can now apply Lemma 5 to show the desired result. This result indicates

that any policy from the threshold set PT improves performance relative to the benchmark case

of no flow control in terms of the expected number of customers conditional on the system being

congested. For example, in managing an emergency department, this implies that the number of

patients beyond the safe threshold conditional on being overcrowded can be reduced by imple-

menting a flow control policy with threshold structure. Moreover, the difference is greater than

the difference between the expected number of customers under the benchmark policy O and the

expected number of customers under any policy from threshold policy set PT . Hence, for cases

when the expected congestion in overcrowded state is of high managerial concern, the improvement

in the main domain of interest is even greater than the improvement over the whole domain.

5. Conclusion

Motivated by the situations where system performance is critical in the tail, we develop a new

approach for comparing managerial policies: new stochastic orderings, K-level stochastic order and

K-convex stochastic order, that focuses solely on the tail of the distribution and a new performance

21

measure that again, focuses on the expectation in the tail. For analytical convenience, we introduce

a transformation technique, head-uniform transformation, that uniformizes the mass in the head

of the distribution and hence lets us ignore the impact of policies on the performance in the head

of the distribution.

We demonstrate the merit of our performance measurement technique by analyzing the impact of

threshold-based flow control policies on the performance of queueing systems with multiple multi-

server nodes. We first show that such policies do not show improvement according to the stochastic

ordering in the usual sense and convex ordering defined in the existing literature. However, using

our new approach, we show that such policies improve the performance in the tail. Our method

is a powerful decision-making tool that can be applied in cases where the tail performance is of

highest managerial concern and can be used to identify policies that perform well according to

many practical performance measures that focus on the right tail of the distribution.

6. Appendix

Parameters λi, µi, and Mi are specific to each queueing node i; similarly, stationary probabilities

πi are specific for each queueing node i. In the proofs below, we suppress subscript i for ease of

readability, the results hold for all i.

Proof of Lemma 1 This proof is similar to the proof of the first stochastic order in usual sense

outlined in Muller and Stoyan (2002).

Definition 1 is equivalent to:

E [IU (X)]≤E [IU (Y )] ∀ upper set U ⊆ [K,∞).

From Real Analysis (Royden and Fitzpatrick 2010), a function can be expressed as a limit of simple

functions, and a non-decreasing function as fK+

(X) in the domain [K,∞) can be expressed as:

fK+

(X) =st limm→∞

m∑j=1

cj IUj (X)− d

for some {Cj}mj=1≥ 0 and {Uj}mj=1

nested, i.e., Uk ⊃ Ul for k<l.

22

And hence, EK+[f (X)], if exists, can be calculated as

EK+

[f (X)] =E[fK

+

(X)]

=E

[limm→∞

m∑j=1

cj IUi (X)− d

]= lim

m→∞E

[m∑j=1

cjIUj (X)− d

],

where Uj ⊆ [K,∞).

The third equality is due to the Lebesgue’s monotone convergence theorem. Thus, we have:

X≤K−stY ⇐⇒EK+ [f (X)]≤EK+ [f (Y )] .

Next, we show that

X≤K−stY ⇐⇒ FX (t)≤ F Y (t) ∀t≥K ⇐⇒ UK(X)≤st UK(Y ).

Note that: F UK(X) (t) = FX (t) ∀t≥K, so

FX (t)≤ F Y (t) ∀ t≥K ⇐⇒ F UK(X) (t)≤ F UK(X) (t) ∀ t≥K.

Since FX (K) = FUK(X) (K)≥ FY (K) = FUK(Y ) (K), we have ∀ t≤K:

fUK(X) (t) =FUK(X) (K)

K≥ fUK(Y ) (t) =

FUK(Y ) (K)

K.

This implies:

F UK(X) (t)≤ F UK(X) (t) ∀ t≤K.

And hence (combined with the above),

UK(X)≤st UK(Y ).

The opposite direction of proof is obvious:

UK (X)≤st UK (Y ) ⇔ F UK(X) (t)≤ F UK(X) (t) ∀ t ⇒

F UK(X) (t)≤ F UK(X) (t) ∀ t≥K ⇐⇒ FX (t)≤ F Y (t) ∀ t≥K. �

23

Proof of Lemma 2 For ease of readability, we suppress subscript i in the consequent analysis.

Rate conservation condition 1 implies:

E[λ(NP)]

=∞∑k=0

πP (k)λP (k) = λ0

πP (0)

λP (0) +M∑m=1

m−1∏k=0

λP (k)

m!µmλP (m) +

∞∑m=1

M−1∏k=0

λP (k)

M !µM

m−1∏k=0

λP (M + k)

(Mµ)m λP (M +m)

= λ0 (18)

Dividing both sides of Equation 18 by Mµ we have:

πP (0)

λP (0)

Mµ+

M∑m=1

m−1∏k=0

λP (k)

m!µmλP (m)

Mµ+∞∑m=1

M−1∏k=0

λP (k)

M !µM

m−1∏k=0

λP (M + k)

(Mµ)m

λP (M +m)

Mµ

=λ0

Mµ= ρ0.

We now work on the left-hand side (LHS). Bringing λP(m) and λP(M +m) terms in:

πP (0)

λP (0)

µ

1

M+

M∑m=1

m∏k=0

λP (k)

(m+ 1)!µm+1

m+ 1

M+∞∑m=1

M−1∏k=0

λP (k)

M !µM

m∏k=0

λP (M + k)

(Mµ)m+1

Separating out the M-th term from the summation in the second term and adding it to the last

term, we get:

πP (0)

λP (0)

µ

1

M+M−1∑m=1

m∏k=0

λP (k)

(m+ 1)!µm+1

m+ 1

M

+

M∏k=0

λP (k)

(M + 1)!µM+1

M + 1

M+∞∑m=1

M−1∏k=0

λP (k)

M !µM

m∏k=0

λP (M + k)

(Mµ)m+1

Note that

M∏k=0

λP (k)

(M+1)!µM+1M+1M

=

M−1∏k=0

λP (k)

M !µMλP (M)

Mµ, thus, replacing m+1 by m and changing the ranges

accordingly, we re-write LHS as:

πP (0)

M∑m=1

m−1∏k=0

λP (k)

m!µmm

M+∞∑m=1

M−1∏k=0

λP (k)

M !µM

m−1∏k=0

λP (M + k)

(Mµ)m

= ρ0 (19)

We will use equation 19 for convenience in the paper.

Proof of Lemma 3 Let

Aj =1−FNj (M)

πj (0)=FNj (M)

πj (0)(20)

24

and

Bj =M∑m=1

m−1∏k=0

λj (k)

m!µmm

M(21)

From the normalization condition we get:

∞∑k=0

πP (k) = 1

πP (0)

1 +M∑m=1

m−1∏k=0

πP (k)

m!µm+∞∑m=1

M−1∏k=0

λP (k)

M !µM

m−1∏k=0

λP (M + k)

(Mµ)m

= 1

The normalization condition and equation 19 from Lemma 2 can be rewritten:

πP (0)

[1

πP (0)FNP (M) +AP

]= 1 (22)

πP (0)[BP +AP

]= ρ (23)

Multiplying Equation 22 by ρ and comparing with Equation23, we get

ρ

πP (0)FNP (M) + ρAP =BP +AP

And hence,

AP =

ρ

πP (0)FNP (M)−BP

1− ρ=

ρ

1− ρFNP (M)

πP (0)− 1

1− ρBP (24)

Plugging 24 into 23, we get:

πP (0)

[BP +

ρ

πP (0)FNP (M)−BP

1− ρ

]= πP (0)

[ρ

1− ρ

(FNP (M)

πP (0)−BP

)]= ρ ⇒

πP (0)

(FNP (M)

πP (0)−BP

)= 1− ρ

πP (0)

1 +M∑m=1

m−1∏k=0

λP (k)

m!µm−

M∑m=1

m−1∏k=0

λP (k)

m!µmm

M

= πP (0)

1 +M∑m=1

m−1∏k=0

λP (k)

m!µmM −mM

= 1− ρ

By the definition of our policy set PT , λP (k)≥ λ0 ∀ k ≤ S, where at least one inequality is

strict. Then,

M∑m=1

m−1∏k=0

λP (k)

m!µmM −mM

>M∑m=1

(λ0)m

m!µmM −mM

Hence, πP (0)<πO (0). �

25

Proof of Lemma 4 We will show that πPi (S)>πOi (S) ∀ P ∈PT by contradiction:

Define a differential function for a policy P as follows:

hP (k)def= πP (k+ 1)− λ0

µmin(k+ 1,M)πP (k) k≥ 1.

We can see that:

hO (k) = 0 ∀ khP (k)≥ 0 ∀ k≤ S

hP (k)≤ 0 o/w

(25)

Suppose πP (S)<πO (S), then πP (k)≤ πP (k) ∀ k≤ S, where at least one inequality is strict, since

hP (k)≥ 0 ∀ k≤ S with at least one strict inequality.

On the other hand, πP (k)≤ πO (k) ∀k > S since hP (k)≤ 0.

So, we must have:∞∑k=0

πP (k)<∞∑k=0

πO (k) = 1

Which is a contradiction. �

Proof of Theorem 1 Step 1. We will first show that

1−FNP (M)

FNP (M)<

1−FNO (M)

FNO (M)

Using definitions 20 and 21, we can express1−F

Nj(M)

FNj

(M)as:

1−FNj (M)

FNj (M)=

Aj

FNj

(M)

πj(0)

=

(ρ

1−ρFNj

(M)

πj(0)− 1

1−ρBj)

FNj

(M)

πj(0)

=ρ

1− ρ− 1

1− ρBj

FNj

(M)

πj(0)

Now, we want to show that BPFNP (M)

πP (0)

> BOFNO (M)

πO(0)

which is equivalent to

M∑m=1

m−1∏k=0

λP (k)

m!µmmM

1 +∑M

m=1

m−1∏k=0

λP (k)

m!µm

>

∑M

m=1

(λ0)m

m!µmmM

1 +∑M

m=1

(λ0)m

m!µm

↔

26

M∑m=1

m−1∏k=0

λP (k)

m!µmm

M+

M∑m=1

m−1∏k=0

λP (k)

m!µmm

M

M∑m=1

(λ0)m

m!µm>

M∑m=1

(λ0)m

m!µmm

M+

M∑m=1

m−1∏k=0

λP (k)

m!µm

M∑m=1

(λ0)m

m!µmm

M

M∑m=1

m−1∏k=0

λP (k)

m!µmm−

M∑m=1

(λ0)m

m!µmm

+

M∑m=1

m−1∏k=0

λP (k)

m!µmm

M∑m=1

(λ0)m

m!µm−

(M∑m=1

(λ0)m

m!µmm

) M∑m=1

m−1∏k=0

λP (k)

m!µm

> 0

M∑m=1

m−1∏k=0

λP (k)

m!µmm−

M∑m=1

(λ0)m

m!µmm

+

(M∑m=1

(λ0)m

m!µm

) M∑m=1

m−1∏k=0

λP (k)

m!µmm−

M∑m=1

(λ0)m

m!µmm

+

+

(M∑m=1

(λ0)m

m!µmm

) M∑m=1

(λ0)m

m!µm−

M∑m=1

m−1∏k=0

λP (k)

m!µm

> 0

(1 +

M∑m=1

(λ0)m

m!µm

) M∑m=1

m−1∏k=0

λP (k)

m!µmm−

M∑m=1

(λ0)m

m!µmm

−(

M∑m=1

(λ0)m

m!µmm

) M∑m=1

m−1∏k=0

λP (k)

m!µm−

M∑m=1

(λ0)m

m!µm

> 0

We use the following notations as matter of convenience

amdef=

(λ0)m

m!µm; bm

def=

m−1∏k=0

λP (k)

m!µm

The above inequality can be expressed:(1 +

M∑m=1

am

)(M∑m=1

(bm− am)m

)−

(M∑m=1

amm

)(M∑m=1

(bm− am)

)> 0

For M = 1, the inequality is trivially true.

For M ≥ 2:(1 +

M∑m=1

am

)(M∑m=1

(bm− am) +M∑m=2

(bm− am) (m− 1)

)−

(M∑m=1

am +M∑m=2

am (m− 1)

)(M∑m=1

(bm− am)

)> 0

M∑m=2

(bm− am) (m− 1)+M∑m=1

(bm− am)+

(M∑l=1

al

)M∑m=2

(bm− am) (m− 1)−

(M∑m=2

am (m− 1)

)(M∑m=1

(bm− am)

)> 0

M∑m=1

(bm− am)m+

(M∑l=1

al

)(M∑m=2

(bm− am) (m− 1)

)−

(M∑m=2

am (m− 1)

)(M∑m=1

(bm− am)

)> 0

27

We will show this inequality using induction on M :(M∑l=1

al

)(M∑m=2

(bm− am) (m− 1)

)−

(M∑m=2

am (m− 1)

)(M∑m=1

(bm− am)

)> 0

Obviously, it holds for M = 1.

Suppose it holds for M = 1, . . . ,K,(K∑l=1

al

)(K∑m=2

(bm− am) (m− 1)

)−

(K∑m=2

am (m− 1)

)(K∑m=1

(bm− am)

)> 0

we will demonstrate it for M =K + 1. Expanding the left-hand side, we get:(K+1∑l=1

al

)(K+1∑m=2

(bm− am) (m− 1)

)−

(K+1∑m=2

am (m− 1)

)(K+1∑m=1

(bm− am)

)=

=

(aK+1 +

K∑l=1

al

)(K (bK+1− aK+1) +

K∑m=2

(bm− am) (m− 1)

)−

(KaK+1 +

K∑m=2

am (m− 1)

)((bK+1− aK+1) +

K∑m=1

(bm− am)

)=

=

(K∑l=1

al

)(K∑m=2

(bm− am) (m− 1)

)+

(K∑l=1

al

)K (bK+1− aK+1) + aK+1

K∑m=2

(bm− am) (m− 1)+

aK+1K (bK+1− aK+1)−

(K∑m=2

am (m− 1)

)(K∑m=1

(bm− am)

)−

(K∑m=2

am (m− 1)

)(bK+1− aK+1)−

KaK+1

K∑m=1

(bm− am)−KaK+1 (bK+1− aK+1) =

=

(K∑l=1

al

)(K∑m=2

(bm− am) (m− 1)

)−

(K∑m=2

am (m− 1)

)(K∑m=1

(bm− am)

)+

K

(K∑l=1

al

)(bK+1− aK+1) + aK+1

K∑m=2

(bm− am) (m− 1)−

KaK+1

K∑m=1

(bm− am)−

(K∑m=2

am (m− 1)

)(bK+1− aK+1)

By induction hypothesis, the first two terms produce a positive result. We will show the rest is

positive:

KK∑l=1

al (bK+1− aK+1) + aK+1

K∑m=2

(bm− am) (m− 1)−KaK+1

K∑m=1

(bm− am)−K∑m=2

am (m− 1) (bK+1− aK+1) =

28

=KbK+1

K∑l=1

al−KaK+1

K∑l=1

al + aK+1

K∑m=2

bm (m− 1)− aK+1

K∑m=2

am (m− 1)−

KaK+1

K∑m=1

bm +KaK+1

K∑m=1

am− bK+1

K∑m=2

am (m− 1) + aK+1

K∑m=2

am (m− 1) =

=KbK+1

K∑l=1

al + aK+1

K∑m=2

bm (m− 1)−KaK+1

K∑m=1

bm− bK+1

K∑m=2

am (m− 1) =

=

[KbK+1

K∑l=1

al− bK+1

K∑m=2

am (m− 1)

]−

[KaK+1

K∑m=1

bm− aK+1

K∑m=2

bm (m− 1)

]=

= bK+1

K∑l=1

al (K − l+ 1)− aK+1

K∑l=1

bl (K − l+ 1) =K∑l=1

(bK+1al− aK+1bl) (K − l+ 1)

Consider

bK+1al−aK+1bl =

K∏k=0

λP (k)

(K + 1)!µK+1

(λ0)l

l!µl− (λ0)

K+1

(K + 1)!µK+1

l−1∏k=0

λP (k)

l!µl=

(λ0)ll−1∏k=0

λP (k)

(K∏k=l

λP (k)− (λ0)K−l+1

)(K + 1)!µK+1l!µl

For all S ≥M and P ∈PT we have: λP (k)≥ λ0 ∀ k≤ S hence,K∏k=l

λP (k)− (λ0)K−l+1

> 0, which

completes the proof for the inequality: BPFNP (M)

πP (0)

> BOFNO (M)

πO(0)

.

We now have:

1−FNP (M)

FNP (M)=

ρ

1− ρ− 1

1− ρBP

FNP (M)

πP (0)

<ρ

1− ρ− 1

1− ρBO

FNO (M)

πO(0)

=1−FNO (M)

FNO (M)

Which is equivalent to:

FNO (M)−FNP (M)FNO (M)<FNP (M)−FNP (M)FNO (M)⇐⇒ FNO (M)<FNP (M)

Step 2. Now, we show that

FNO (S)<FNP (S) ∀ S ≥M :

Next, we show that πP (M)> πO (M) by contradiction. Suppose πP (M)< πO (M), from the dif-

ferential function 25 and the inequality πP (0)<πO (0), we have that πP (t)<πO (t) ∀ t≤M . This

indicates that FNP (M)<FNO (M), contradicting the above result.

Finally, from the differential function (Equation 25), πP (t) > πO (t) ∀ M ≤ t ≤ S ∀ P ∈ PT .

Resulting in

FNO (S)<FNP (S) ∀ S ≥M. �

29

Proof of Theorem 2 As shown in Theorem 1 and Lemma 1 above, NP <S−st NO is equivalent

to US(NP)<st US(NO).

Using the result from Do and Iyer (2015), US(NP) <st US(NO) implies T (US(NP)) <st

T (US(NO)).

By Property 1 of the T -transform, the mean is shifted by 12

under the transformation, and hence:

E[T (US(NO))]−E[T (US(NP))] =E[US(NO)]−E[US(NP)] = ∆P > 0.

Since the pdf of T (US(NP)) starts from above and crosses down only once the pdf graph of

T (US(NO)), following similar steps in Do and Iyer (2015), we can establish:

T (US(NP)) + ∆P <cx T (US(NO))∀ P ∈PT .

Using Property 2 of the T -transform, this implies:

US(T (NP)) + ∆P = US(T (NP) + ∆P)<cx US(T (NO))∀ P ∈PT .

By Definition 3, we then have

T (NP) + ∆P <S−cx T (NO)∀ P ∈PT . �

Proofs of Lemma 5 and Proposition 1 follow directly from the steps outlined in the main body

of the paper and hence, are omitted from the Appendix.

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