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Deployment-to-dwell metrics and

supply-based force sustainment

Sarah E. Evans

US Air Force Special Operations Command, Hurlburt Field, Florida, USA, and

Gregory Steeger

US Air Force Special Operations Command, USAFA, Colorado, USA

Abstract

Purpose –In the present fast-paced and globalized age of war, special operations forces have a comparative

advantage over conventional forces because of their small, highly-skilled units. Largely because of these

characteristics, special operations forces spend a disproportionate amount of time deployed. The amount of

time spent deployed affects service member’s quality of life and their level of preparedness for the full

spectrum of military operations. In this paper, the authors ask the following question: How many force

packages are required to sustain a deployed force package, while maintaining predetermined combat-

readiness and quality-of-life standards?

Design/methodology/approach –The authors begin by developing standardized deployment-to-dwell

metrics to assess the effects of deployments on service members’quality of life and combat readiness. Next,

they model deployment cycles using continuous time Markov chains and derive closed-form equations that

relate the amount of time spent deployed versus at home station, rotation length, transition time and the total

force size.

Findings –The expressions yield the total force size required to sustain a deployed capability.

Originality/value –Finally, the authors apply the method to the US Air Force Special Operations

Command. This research has important implications for the force-structure logistics of any military force.

Keywords Readiness, Continuous time Markov chains (CTMC), Deployment-to-dwell (D2D),

Force sustainment, Personnel tempo (PERSTEMPO), Special operations forces (SOF)

Paper type Research paper

1. Introduction

When embarking upon any long-term endeavor, it is important to count or at least estimate

the cost necessary to ensure completion. The trouble with counting the cost of wars is that

the path to “completion”is often fraught with unknowns. Furthermore, the business of war

itself is constantly changing (Votel et al.,2016). Therefore, warﬁghters and planners must do

their best, despite all the unknowns, to ensure the provided military capability is sustainable

with the resources on hand.

Special operations forces (SOFs) are uniquely capable to adapt to the changing rigors of

war. SOFs are small highly skilled units designed to respond quickly to emerging crises in

any part of the world, but they are not designed to conduct a major combat operation

without conventional forces (Bucci, 2015;Spulak, 2007). As evidenced by recent global

conﬂicts, SOF use has increased, perhaps to the detriment of their future use (Hennigan,

© In accordance with section 105 of the US Copyright Act, this work has been produced by a US

government employee and shall be considered a public domain work, as copyright protection is not

available. Published in Journal of Defense Analytics and Logistics. Published by Emerald Publishing

Limited.

JDAL

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Received 22May 2017

Revised 3 October2017

28 February 2018

29 March 2018

Accepted 4 April2018

Journal of Defense Analytics and

Logistics

Vol. 2 No. 1, 2018

pp. 2-21

Emerald Publishing Limited

2399-6439

DOI 10.1108/JDAL-05-2017-0009

The current issue and full text archive of this journal is available on Emerald Insight at:

www.emeraldinsight.com/2399-6439.htm

2017;Robinson, 2013;Watson, 2017). Employing SOF in a manner that puts production

capacity in peril leads to decreasing capability (JP3-05, 2014). Logically, identifying and

studying examples of decreasing capability can help reduce the risk of improper

employment. Additionally, depleting capability negatively impacts members’combat

readiness and quality of life (Losey, 2017b;Woody, 2017).

For the purposes of this paper, an individual’s“quality of life”is based on a multitude of

factors including mental, physical, social and spiritual well-being. Combat readiness and

quality of life are related because physical and mental well-being affect ability to deploy, in

addition to job-related competencies and regular training (Tucker et al.,2005;Rounds, 2010;

Szivak and Kraemer, 2015). Consequently, training, deployment and recovery compose the

readiness cycle.

One way to measure the readiness and quality of life of an individual or an organization

is to look at deployment-to-dwell (D2D) ratios (Dabkowski et al.,2009;Langstroth, 2013;

MacGregor et al., 2014;Trautmann et al.,2015). A D2D ratio is the amount of time an

individual (or group) is operationally deployed to the time the individual (or group) is not

deployed (PM 15-37, 2016). D2D ratios are typically normalized to the length of

a deployment so that they are reported as 1:days in dwell divided by days away on the

deployment. The recent necessity to obtain waivers for breaching D2D thresholds, at the US

Secretary of Defense (SecDef) level, along with low D2D ratios are possible indications of

SOF’s depleting readiness capacity (Losey, 2017a). These occurrences motivate the

overarching research question: How many force packages are needed to sustain a deployed

force package, while maintaining predetermined combat-readiness and quality-of-life

standards?

As early as May 2005, in guidance on Global Force Management, the SecDef expressed

concerns about US military forces’operations tempos and their impacts on the troops

(SecDef, 2007;USD, 2005;Chamberlain et al.,2005). In this context, operations tempo refers

to how often an individual is away from home because of combat-related deployments or

temporary duties. Later, SecDef (2007) established two metrics to measure an individual’s

operations tempo: personnel tempo and D2D. Personnel tempo measures an individual’s

operations tempo based on total days away from home station for any duty-related purpose,

whereas D2D measures an individual’s operations tempo based solely on combat-related

deployments. An operational deployment begins when a member departs his or her home

station, or en route training location, to meet a SecDef-approved operational requirement

(PM 15-37, 2016). An operational deployment ends when the individual arrives back at his or

her home station (PM 15-37, 2016). The SecDef imposed restrictions on force supply, for both

active and reserve forces, by establishing both personnel tempo and D2D goals and

limitations (SecDef, 2007). As originally stated, “the planning objective for the Active Force

remains one year deployed to two years at home station”(i.e. 1:2 D2D). The SecDef directive

also required that members maintain D2D ratios above 1:1. Though this policy was set

nearly 10 years ago because of insatiable demand and little to no enforcement of supply

restrictions, SOFs have operated at close to 1:1 D2D ratios (Copp, 2018;Losey, 2018).

There are many debates on the scope of SOF’s roles and responsibilities (Robinson, 2013).

SOFs are faster and more ﬂexible than conventional forces, and SOFs have the skills to

address almost any mission. However, it is important to realize that the very characteristic

that gives SOF the ability to do what SOF does is the very thing that limits how much they

can do: their small size. Many SOF capabilities worldwide are “low density high demand”or,

in other words, they are limited assets or forces with unique mission capabilities stressed by

continual high requirements for their capabilities. The reality is that as a low-density high-

demand force, SOF cannot do everything demanded because their resources are limited.

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force

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3

Therefore, it is in the best interest of the organization to objectively determine the resources

that are needed to sustain enduring deployment requirements.

In a resource-constrained environment, protecting production capacity is paramount. In

this context, protecting production capacity means producing a sustainable capability by

ensuring the availability of training, resources and equipment. Arguably, the most valuable

resource is personnel. To protect production capacity in this environment, with less than 100

per cent manning, policies must be put in place to sustain supply. Even with manning at 100

per cent, there can still be readiness problems, especially in the context of insatiable demand.

Typically, SOF responds to increasing demands by increasing the quantity supplied. For

instance, for US SOF, it is okay to break the SecDef mandate to maintain a D2D of at least

1:1 so long as those breaking the mandate are volunteers and have approval from the ﬁrst

ﬂag ofﬁcer in their chain of command (PM 14-07, 2014). As a result, supply has increased

with demand. An illustration of this paradigm, commonly known as perfectly elastic supply

in economics, is provided in Figure 1(a).

In general, if supply exceeds demand and there are no other limiting factors in the

system, the number of transactions in the system are limited by the demand. Conversely, a

system with insatiable demand, but constrained supply, is limited to the number of

transactions possible given the existing supply. In actuality, SOF is ultimately a system

Figure 1.

(a) Perfectly elastic

supply; (b) perfectly

inelastic supply; and

(c) SOF supply

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with limited supply under insatiable demands.In economics, this situation is called perfectly

inelastic supply and is shown in Figure 1(b). In this case, the price could be viewed as

opportunity cost or risk because both increase as demand increases, though the quantity

supplied remains ﬁxed. To prioritize one operation over another, in the context of limited

supply, is to do that operation at the opportunity cost of the other. Additionally, as there is

more unfulﬁlled demand for SOF, risk increases in the ﬁeld. Structuring the forces in a

manner, which protects production capacity, becomes all the more important in a resource-

constrained environment because the parties making demands are not necessarily affected

by the costs, either in the long or short term. Rather the service members assume the costs as

their quality of life and combat readiness are affected. Therefore, to account for supply

limitations, the capabilities should be presented as in Figure 1(c).Figure 1(c) demonstrates

how capability supply ceilings allow increases in demand to a predetermined point, after

which all increases in demand have no effect on the quantity supplied.

This paper presents a supply-based model for determining the required force strength

necessary to sustain an enduring war-ﬁghting capability. We model the problem using a

continuous time Markov chain (CTMC) and use the chain’s limiting behavior to determine

steady-state equations for D2D ratios. The equations yield the relationship between the force

multiplier (i.e. the number of identical force packages used to sustain one that is deployed),

deployment rotation length, transition time and non-availability of forces.

This research is valuable for several reasons. First, it standardizes an objective

measurement of readiness by clearly deﬁning D2D. Second, it develops a method that

enables a decision-maker to efﬁciently allocate resources, based on existing supply, while

preserving readiness for the long term. Finally, to mathematically justify current and future

force employment decisions, this research derives an equation that relates D2D to key

deployment planning factors. Our work is unique and ﬁlls a gap in the literature because, to

our knowledge, we are the ﬁrst to relate force sustainment to a quality-of-life metric. This

work has undeniable applications to and implications for SOF worldwide and military

forces in general.

The remainder of this paper is organized as follows: Section 2 provides the necessary

background and framework for the problem. Next, Section 3 logically frames and then

presents the methodology in general terms. Section 4 applies the method to US Air Force

Special Operations Command’s (AFSOC’s) active duty forces. Following this, Section 5

states the ﬁndings and recommendations.

2. Background

Before attempting to answer the overarching research question, it is essential to properly

frame the problem. Any attempt to structure forces, whether conventional or SOF, should

begin with determining the capability that is needed. Properly sizing and structuring each

force package, or all the equipment and personnel associated with a deﬁned military

capability, is important for creating measurable capabilities and reasonable expectations.

2.1 The big picture: capability-based force structuring

Figure 2 explains how the force should be built and structured based on supply. Three

phases are used to describe the process: building the force, employing or deploying the force

and sustaining the force. In each of the phases, answers to the questions shown in the ﬁgure

need to be determined.

The ﬁrst step in the process is to determine the capability or capabilities to be provided.

It is important to understand that larger force packages gain maintenance efﬁciencies.

However, the tradeoff for these efﬁciencies manifests itself in decreased ﬂexibility. Smaller

Supply-based

force

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5

force packages allow for enhanced projection of the capability to more locations. When

creating a force package, it is important to make its capability objectively measurable to

effectively communicate the capability’s readiness.

We assume initiation in the third phase of the process; an appropriately sized force

package has been built and employed, and now it is necessary to determine the number of

identical force packages required to sustain each one that is deployed long term. To sustain

the force package, we must ﬁrst determine how to quantify force sustainment. Force

sustainment is achieved when force packages are used to provide an enduring capability

and that all the personnel involved are healthy. We measure personnel health via D2D.

2.2 Measuring and standardizing operations tempo

The D2D metric we develop meets all of the standards listed in Harrison (2014). The D2D

metric measures outputs rather than inputs, is linked to strategy, is quantiﬁable and avoids

subjective assessments. D2D metrics can be used in two separate ways to measure either the

historical health or the combat capability (or availability) of the force. We focus on the

former.

Based on the deﬁnitions provided for dwell and an operational deployment (Section 1;

PM 15-37, 2016), individual historic D2D ratios can only be computed for individuals with at

least two deployments. To be counted, a dwell period must be bookended with an

operational deployment return date and an operational deployment departure date. Each

individual’s historic D2D ratio is based solely on the length of their penultimate operational

deployment and the length of the dwell period immediately following it. Of the three

individuals shown in Figure 3, a historic D2D ratio can only be computed for X. X’s historic

D2D ratio of 1:2 is based on their deployment from 1 January to 31 March and their dwell

period from 1 April to 30 September. A historic D2D ratio cannot be computed for Y because

Figure 2.

Force structure

overview

Figure 3.

Historic D2D example

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they have not departed on their second deployment and, consequently, their dwell period

has not ended. Finally, a historic D2D ratio cannot be computed for Z because they are yet to

start their ﬁrst dwell period.

Though D2D ratios are individual metrics, a group’s average historic D2D ratio can be

used to answer health-of-the-force questions for different groups of individuals. In an effort

to capture the most representative metric, a group’s average historic D2D ratio is computed

holistically using historic D2D ratios, when they can be computed, from everyone assigned

to the group. The holistic average is used to place less emphasis on individual historic D2D

ratios that may be considered outliers; an individual with a short deployment followed by a

long period of dwell may skew the group’s average. To compute the holistic average dwell

ratio, one divides the sum of the dwell lengths (in days) for all the members in the group, for

which a D2D ratio can be computed, by the deployment lengths (in days) for all the members

in the group, or mathematically (for npersonnel with valid D2D ratios in a group):

average historic dwell ¼dwell1þdwell2þ... þdwelln

deploy1þdeploy2þ... þdeployn

:

The average historic D2D is then reported as 1: average historic dwell.

An example showing how to compute a group’s average historic D2D ratio is given in

Table I. Assume that the entire group consists of two members, A and B, with penultimate

deployment and last dwell lengths shown. Weighting each deployment equally, the group’s

average historic D2D ratio is 1: 1þ3

1þ1

¼1: 2:In the holistic average, rather than weighting

each individual’s D2D ratio equally, the individual ratios are weighted based on their

lengths. The group’s holistic historic D2D ratio is 1: 60 þ90

60 þ30

¼1: 1:67:Having developed a

standardized way to measure the health of the force, this paper now reviews force

sustainment models, both in practice and in the academic literature.

2.3 Force sustainment approaches

Each of the USA’s services has a unique way of sustaining their forces via readiness cycles.

The intent of each plan is to rotate personnel and equipment in such a way that training,

quality of life, maintenance and deployment requirements are all met. Table II describes the

methods of the US Air Force, Army, Navy and USSOCOM. As shown in Table II, none of

these plans are functioning as perfectly as intended.

2.4 Literature review

More often than not, work in this area views force sustainment from the perspective of

readiness measurement or manpower requirements. There is a wide variety of approaches

to measuring readiness (Harrison, 2014;Freeman et al., 2014;Barzily et al.,1979;Scales et al.,

2011). Our review of the literature on personnel modeling is summarized in Table III.Note

that the literature is sorted by method and then by publication date. In an effort to be

concise, we only discuss a sample of the literature from each of the methods used and focus

Table I.

Example historic

D2D calculation

Deployment length (in days) Dwell length (in days) Individual D2D ratio

A 60 60 1:1

B 30 90 1:3

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on the literature that is closely related to our work. More comprehensive literature reviews

and surveys can be found in the studies of Gass (1991),Wang (2005),Guerry and Feyter

(2009) and Parlier (2016).

Kinstler et al. (2008) use a Markov model to rectify rank imbalances in the Navy Nurse

Corps. Filinkov et al. (2011) create a software tool to test the personnel sustainability of a

land force structure in terms of career progression and operational considerations for the

Australian Army. Richmond et al. (2012) model the population of ground forces to manage

personnel and major system sustainability with Markov techniques for the Australian

Army. Zais and Zhang (2016) examine stay or leave decisions in the US Army using a

Markov chain model. Mitchell (1993) uses a simulation to estimate the impact on training

requirements of force structure decisions for the US Air Force. Pall et al. (2007) use a

simulation to examine personnel and materiel policies for the Canadian Army. Kim et al.

(2012) applied a stochastic optimization model to manage the uncertainty of demand and

supply for knowledge workers. Durbin and Wright (1967) use linear programing to manage

overseas tour lengths for high-demand positions in the US Air Force. Whitney et al. (2013)

consider the effects of force organization on capability fulﬁllment using a qualitative

methodology for the Australian Army.

While the literature above has important implications, many are impractical for

informing the day-to-day decision-making that impacts lives, such as how many force

packages to deploy. The effects of military organization decisions on the lives of the people

which compose it are important considerations for force management because people’slives

are fundamentally entwined with their combat readiness (AFI90-506, 2014). While studies

show that deployments have both positive and negative effects on retention, a reoccurring

top concern for service members is the amount of time they spend separated from their

families (Fricker, 2002;Badger, 2004). Additionally, deployment duration has been

Table II.

Deployment cycle

plan comparison

Service/

command Nomenclature Directive Notable characteristics

Air Force Air

Expeditionary

Force Cycle

Air Force Instruction

10-244

3-phase, demand-driven cycle. Surge above 12

months may require signiﬁcant actions to

reconstitute the force (AFI10-244, 2002).

Disproportionate deployment burdens have

been a problem (Losey, 2016)

Army Army Force

Generation

Army Regulation

525-29

3-phase, demand-driven cycle. Will be replaced

by Sustainable Readiness Model in ﬁscal year

2017 because of increased demand in

conjunction with decreasing resources (Army

Readiness Guidance, 2017)

Navy Optimized

Fleet Response

Plan

Ofﬁce of the Chief of

Naval Operations

Instruction 3000.15A

4-phase, supply-driven cycle. Maintains the

capacity to rapidly increase forward presence as

world events dictate and additional funding

becomes available (OPNAV Instruction

3000.15A, 2014). Of the ﬂeets converted to

OFRP, many have had difﬁculty maintaining

the cycle schedule because of maintenance

overrun (GAO, 2016)

USSOCOM Special

Operations

Force

Generation

USSOCOM Directive

5225-10

3-phase, demand-driven cycle. More of a

guideline than a policy because each SOF

component must also answer to their respective

service (USSOCOM Directive 5225-10, 2013)

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Table III.

Academic literature

on personnel

modeling

Reference Research question Method Application

Mitropoulos (1983) What metrics best describe members’

professional evolution in hierarchical

organizations?

Markov model General

Weigel and Wilcox

(1993)

How do high-level personnel planning

decisions impact troops at the

occupational specialty level?

Markov, network, linear

programming and goal-

programming models

US Army

Georgiou and

Tsantas (2002)

What is the optimal way to minimize

cost, in a k-classed hierarchical system,

while meeting workforce demand and

satisfying government constraints and

regulations?

Markov model European Union/

Workforce

Kinstler et al. (2008) How do different policies impact

balance in the rank structure and

retention?

Markov model US Navy Nurse Corps

Filinkov et al. (2011) What force strength is required to meet

operational demands, while considering

individuals’career progression?

Markov model with

hierarchical classes

Australian Army

Richmond et al.

(2012)

How will personnel and/or major

systems populations change over time?

Markov model Australian Army

Zais (2014) What are the workforce requirements

based on uncertain demand?

Markov model, simulation U.S Army

Zais and Zhang

(2016)

What incentives have the greatest

impact on personnel retention?

Markov model, stochastic

dynamic programming

US Army

Bender and

Isbrandt (1991)

What are the effects of different policy

options on career progression?

Discrete event simulation Canadian Forces

Mitchell (1993) How many individuals need to be

trained annually, for every occupation?

Discrete event simulation United States Air Force

Zegers and

Isbrandt (2006)

What simulation tool most accurately

depicts military training and career

progression?

Discrete event simulation Canadian Forces

Pall et al. (2007) What is the most efﬁcient way to

provide and sustain qualiﬁed units for

operational tasks?

Discrete event simulation Canadian Army

Moorhead and

Halbrohr (2010)

What simulation tool best determines

the ability of the Canadian Forces to

meet the personnel demands of

operations?

Discrete event simulation Canadian Forces

Cao et al. (2010) How can human capital supply chain

decisions be improved to drive better

business performance with integrated

OR models?

Stochastic modeling and

optimization

Human Capital Supply

Chains

Kim et al. (2012) How does uncertainty of demand for

knowledge services as well as the

supply of knowledge workers impact

recruiting strategy?

Stochastic optimization model Korean Security

Consulting Companies

Durbin and Wright

(1967)

How many personnel are needed to meet

the requirements associated with

rotating personnel between state-side

and overseas locations?

Linear programming United States Air Force

Durso and Donahue

(1995)

What are the impacts of personnel

management policies on the US Army’s

enlisted force?

Decision analysis United States Army

Edwards (1983) What manpower models exist and to

what extent have these been effective in

application?

Qualitative assessment Industry

Whitney et al.

(2013)

What is the Australian Army’s ability

to undertake new or existing

contingencies?

Qualitative assessment Australian Army

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9

associated with negative effects on psychological and physical health (Meadows et al.,2017;

Mulligan et al., 2012;Szivak and Kraemer, 2015). Our work establishes an easy-to-

understand feedback loop for decision-makers which reconnects the costs associated with

these effects on the capability demanded. Additionally, this work ﬁlls a gap in the existing

literature by connecting an objective and mathematically rigorous force sustainment model

to a quality-of-life metric.

3. Methodology

For the intents and purposes of this research, force readiness and quality of life are

measured solely via historic D2D metrics. We use a 1:2 D2D ratio to logically frame and

build the method. After making the necessary logical arguments, we present the resulting

formulas in general terms assuming one wishes to maintain a 1:DD2D ratio, as opposed

to 1:2.

To perpetually sustain one deployed force package and maintain a 1:2 D2D ratio for all of

the associated personnel, at least three force packages are required. The three force

packages will rotate through equal periods of deployment, recovery and preparation

(Figure 4). As they rotate, the D2D ratios for the personnel assigned to each force package

will change as shown. To maintain D2D ratios of 1:2, the preparing forcepackage will not be

ready to deploy until all of its associated personnel have stayed in dwell long enough so that

their individual D2D ratios are 1:2 or better.

The three-force package model assumes 100 per cent manning and availability in all

positions, and instantaneous changeover and transit. In reality, manning shortfalls (in one or

multiple crew positions), manning unavailability (in one or multiple positions) and the

reality of transit time and responsibility changeover make it so the three-force package

model does not provide adequate manpower to maintain 1:2 D2D ratios.

Based on manning shortfalls and non-availability, it may seem reasonable to suggest a

force multiplier of four [Figure 5 (left)].

Figure 5.

Four- force package

model (left) and four-

force package model

with overlap (right)

Figure 4.

Three-force package

model (left) and three-

force package model

accounting for

changeover and

transit time (right)

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There are several problems with suggesting a force multiplier of four based on the

arguments posed thus far. First, the four-force package model compensates for low manning

with additional force package-provided billets. Increasing the force multiplier (i.e. adding

billets) to compensate for manning shortfalls is nonsensical. The focus should be on ﬁlling

the empty billets as opposedto askingfor more billets.

Second, the idea assumes that all the non-available personnel are entirely separate,

or mutually exclusive, from any of the other categories. In reality, non-availability

occurs in all the categories; one may become unavailable just prior to their

deployment, or they may not be able to accomplish the necessary tasks associated

with either the recovery or preparing phases which will create a chain reaction

resulting in the individual not deploying on time later in the cycle. Figure 5 (right)

illustrates how personnel who are not available can also be in one of the other three

categories. Arrows to and from the non-available bucket have been removed to avoid

cluttering the diagram.

Figure 5 (right) still does not address the concern about compensating for low

manning (i.e. empty billets) with additional force package-provided billets.

Consequently, the non-available bucket should only account for the number of non-

available personnel of those assigned, as opposed to those authorized. Even so, Figure 5

(right) suggests that the necessary force multiplier is a number between three and four.

Third, because changeover and transit time are still not considered, incorporating them

is necessary to ensure that there are no manning gaps. In other words, individuals do not

leave the area of responsibility until changeover. The time required for changeover and

transit is an important consideration for low-density high-demand assets because, with

higher operations tempos, it makes deployment periods longer and dwell periods shorter.

For conventional forces, 1:4 D2D rates are more typical so changeover and transit times are

effectively negligible.

Figure 4 (right) shows the resulting D2D ratios using the three-force package model and

assuming 14 days for transit and changeover with a rotation length of 120 days. In this

scenario, the force package’s D2D never reaches 1:2. Based on the arguments posed thus far,

it seems a force multiplier between three and four is necessary to maintain a 1:2 D2D.

However, this conclusion is an impasse; to move forward and better quantify the required

force multiplier, a more sophisticated model is necessary.

Therefore, we ﬁrst model the deployment and dwell periods for the force package

using a CTMC and then compute the steady-state probabilities for each state in the

chain. Figure 6 depicts the transition diagram. The steady-state probabilities are used

to compute historic D2D ratios because they yield the proportion of time a force

package is in dwell versus the proportion of time the force package is deployed. With

an expression for steady-state probabilities, rearranging reveals the dependence of

the force multiplier on rotation length, as well as the amount of changeover and

transit time. This CTMC is a birth-and-death process with two states “deployed”

(state 0) and “in dwell”(state 1).

Figure 6.

Transition diagram

for one force package

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11

To solve for the steady-state probabilities, the transition rates q01 and q10 must be

found. q01 is the rate at which a force package transitions from being deployed to being

indwell.Thenumberofdayseachforcepackageisdeployedisequaltotherotation

length, RL,plustheﬁxed number of changeover and transit time, T.Aforcepackage

transitions from deployed to in dwell once every RL þTdays. In other words,

q01 ¼1

RL þT:

Similarly, q10 is the rate at which a force package transitions from being in dwell to being

deployed. This rate is equivalent to one over the length of time spent in dwell. The length of

time spent in dwell equals the rotation length, RL, times the number of rotations the force

package remains in dwell, minus the ﬁxed number of days spent in changeover and transit,

T. The number of rotations the force package remains in dwell is equal to the number of

force packages that are not deployed (i.e. in dwell) or the force multiplier, fm, minus one.

Changeover and transit days, T, are considered deployment days and, thus, do not count as

dwell. Putting all of this together:

q10 ¼1

RL *fm 1

ðÞ

T:

The steady-state probabilities are derived using the balance equations which, based on the

transition rates, are:

p

0

1

RL þT

¼

p

1

1

RL *fm 1

ðÞ

T

(1)

p

0þ

p

1¼1:(2)

The D2D ratio is now simply stated as:

p

0

p

1

;

which can be expressed by rewriting equation (1) as:

p

0

p

1

¼RL þT

RL *fm 1

ðÞ

T:

Normalizing the length of the deployment and writing D2D in the standard format yields:

D2D¼1: RL þT

RL *fm 1

ðÞ

T:

To maintain a D2D ratio of 1:D:

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RL *fm 1

ðÞ

T

RL þTD:(3)

Finally, to determine how the force multiplier depends on rotation length, as well as

changeover and transit time, solve equation (3) for fm to obtain:

fm Dþ1

ðÞ

RL þT

RL

:(4)

Equation (4) yields the force multiplier necessary to maintain long-term 1: DD2D ratios

based on rotation length, as well as changeover and transit time. Note that the

expression:

RL *fm 1

ðÞ

T

RL þT;

from equation (3) is useful because it yields a force package’s long-term dwell based on

rotation length, a given force multiplier, as well as changeover and transit time. Non-

availability is still not part of the equation.

To account for non-availability of the assigned manpower, divide the number produced

in equation (4) by A, where:

A¼Number Available

Number Assigned :

The ﬁnal relationship that determines the force multiplier needed to maintain 1: DD2D and

accounts for rotation length, changeover, transit time and manpower non-availability is:

fm Dþ1

A

RL þT

RL

:(5)

4. Application

To show the value of this work, our method is applied to the AFSOC. To begin, we discuss

the importance of the holistic D2D average.

Figure 7 depicts individual D2D ratios for members of an AFSOC operations group.

In this particular group, as the histogram on the left shows, approximately 4 per cent of

Airmen have dwell rates greater than 19.5 (i.e. D2D ratios less than 1:19.5). The high dwell

rates skew the average dwell rate signiﬁcantly. In this case, all of the high dwell rates are

based on deployments that lasted less than 30 days. Often times, because of database and

administrative limitations, short trips overseas are coded as deployments when they do not

actually meet the deﬁnition of an operational deployment. Once the short trips are removed,

the mean dwell rate is a more representative metric. However, distinguishing short trips

from actual deployments by removing anything that lasts less than 30 days is completely

subjective because the 30-day cutoff was chosen arbitrarily. In truth, some deployments last

less than 30 days. Typically, when a statistician does not want the measure of central

tendency to be impacted by outliers, they will use the median instead of the mean. However,

outliers (or short deployments) should not be completely ignored but instead should be

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weighted appropriately. The holistic average takes short deployments into account but

weights them according to their length.

Historically, at AFSOC, based on personnel data from the Military Personnel Data

System collected from 2011 through 2014, 85 per cent of authorized billets actually have

personnel assigned and 91 per cent of those assigned are available. Therefore, AFSOC tends

to have 77 per cent of authorized personnel available. Mathematically:

No:Assigned

No:Authorized ¼85%;

No:Available

No:Assigned ¼91%;and No:Available

No:Authorized ¼77%:

As, historically at AFSOC A= 0.91 and as, for AFSOC’s active duty force, the goal is 1:2

D2D ratios, the force multiplier is given as:

fm 3

0:91

RL þT

RL

:

Figure 8 plots the required force multipliers, as a function of rotation length, based on just

the combined changeover and transit time (solid line) and based on the combined

changeover and transit time plus the non-availability of assigned personnel (dashed line).

Note that the required force multiplier decreases, but not linearly, as the rotation length

increases. This inverse relationship is expected, as the number of rotations required

decreases as the rotation length increases and as every rotation requires a ﬁxed amount of

changeover and transit time. As a side note, because each rotation requires a ﬁxed amount

of changeover and transit time, as rotation length increases, the total number of days away

Figure 7.

Left: Dwell rates

histogram (all data)

and right: Dwell rates

histogram

(no deployments

#30 days)

Figure 8.

Force multiplier

needed to maintain

1:2 D2D based on

rotation length (14

days of changeover

and transit time)

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decreases. This relationship creates a tradeoff between rotation length and total number of

days away.

Basedonthemath,Figure 8 plots non-integer-valued force multipliers. Fractions of force

packages are permissible if compensating for personnel non-availability; however, a fraction of

a force package cannot cover a deployment requirement for a full force package. The question

becomes “since it is not possible to deploy fractions of people or crews, is it possible to use

fractions of force multipliers to determine the correct force strength or force package size?”

The answer is, it depends. If the goal is for personnel to maintain an average D2D of 1:2

over multiple deployments and dwell periods, then the answer is yes. If, however, the goal is

to have the manpower necessary so that each dwell period, for each individual, is twice as

long as the deployment preceding it, then the answer is no. The latter requires rounding up

to the next integer-valued force multiplier. The solid line in Figure 8 shows that rounding up

to a force multiplier of four provides the manning necessary to account for changeover,

transit time and non-availability for rotations longer than 90 days. Because of rounding up,

a force multiplier of four will result in individual and average D2D ratios better than 1:2. The

margin by which the D2D ratios are better than 1:2, as shown in Figure 8, increases as

rotation length increases.

To determine how sensitive the results in Figure 8 are to changeover and transit time,

this paper will now examine how the force multiplier is affected by an increase or decrease

in the number of changeover and transit days. Originally the assumption was a total of

fourteen days of changeover and transit time, seven days on either end of each rotation.

Decreasing the number of changeover and transit days to six (three on either end of each

rotation) yields the force multipliers plotted at the bottom of Figure 9. These can be thought

of as a lower bound on the force multiplier (i.e. a best-case scenario). On the other hand,

increasing the number of changeover and transit days to 20 (ten on either end of each

rotation) produces the force multipliers plotted at the top of Figure 9. Ten days of

changeover and transit time on either end of each rotation is sufﬁcient, in most cases, to

place an upper bound on the force multiplier. Note that, for rotations of 60 days or longer, the

force multiplier is generally bounded between three and four.

The key to effectively using the ﬁndings of this research is deﬁning what is meant by

“maintaining a D2D ratio of 1:D.”Here, “maintaining”could mean:

Maintaining an average D2D of 1:Dover all members in a group.

Maintaining an average D2D of 1:D, for each individual in the group, over all of their

individual deployments.

Maintaining a D2D for each individual in the group, after each deployment.

Figure 9.

Sensitivity analysis:

Force multiplier

bounds (6-20 days

changeover and

transit time)

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As this research has shown, the third deﬁnition is the most restrictive and, consequently,

will require the most manpower and the largest force multiplier. However, the third

deﬁnition provides service members with the most predictability and, arguably, the best

quality of life. If the intent is to satisfy either of the ﬁrst two deﬁnitions, then equation (5) is

sufﬁcient to determine the required force multiplier. If, however, the intent is to satisfy the

third deﬁnition, then the results from equation (5) should be rounded up with consideration

for associated risks.

The following example helps to explain some of the ﬁner points of this analysis. Assume

the 123rd Special Operations Group has a requirement to deploy an enduring force package

of ten individuals for 120-day rotations. Also, assume that there are 14 total days of

changeover and transit associated with every rotation, seven days on either end of each

rotation. According to Figure 8, not considering availability, and not rounding up to the

nearest integer, the 123rd Special Operations Group requires a total of 34 personnel.

Figure 10 shows how different groups of personnel will transit and rotate in and out of

theater and dwell over a 494-day period (four full rotations plus changeover and transit

time).

Personnel are gone for 134 days on each deployment; 120 days for the rotation and 14

days for changeover and transit time. However, as there are not four complete force

packages (or groups of ten personnel), dwell periods are neither the same length for each

group nor the same length for each individual after each of their deployments. After their

ﬁrst rotation, Personnel 1-6 are in dwell 226 days after their deployment of 134 days and

have a D2D ratio of 1:1.69. Personnel 7-10 have a longer dwell period of 346 days and,

consequently, a D2D ratio of 1:2.58. Personnel 1-6 have fewer dwell days than Personnel 7-10

because Personnel 1-6 must depart to cover rotation four with Personnel 31-34. The groups

of personnel, as listed in the rows, have alternating D2D ratios of 1:1.69 (shown in grey) and

1:2.58 (shown in white). The average D2D ratio, over all personnel or for each individual,

over multiple complete cycles, is 1:2.

Two main conclusions can be drawn from this analysis and speciﬁcally from the

example above:

(1) Average 1:2 D2D ratios, over multiple individuals or over multiple deployments

for each individual, can be achieved with the force multipliers shown in

Figure 8.

(2) Individual D2D ratios of 1:2 for each individual and following each deployment

cannot be achieved unless the force multipliers shown in Figure 8 are rounded up

to four.

Figure 10.

Example: 123rd

special operations

group changeover,

transit, rotation, and

dwell timeline

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Additionally, note that the example does not account for non-available personnel and,

therefore, is a best-case scenario. When personnel become non-available, variability

within the groups shown in Figure 10 is introduced and the required force multiplier

becomesmoredifﬁcult to compute. Because of non-availability, a force multiplier of

four does not necessarily guarantee that individuals will not drop below 1:2 D2D. This

can happen when someone who is supposed to deploy as part of their normal rotation

becomes non-available just prior to their scheduled departure date, and someone else on

a separate rotation has to ﬁll in and break their 1:2 D2D. If it is unacceptable for

individuals to drop below 1:2 D2D, on such occasions, increasing the force multiplier to

a number larger than four may be justiﬁed. As the preponderance of AFSOC’s rotations

are between 90 and 150 days, and as the goal is to generally maintain individual D2D

ratios of 1:2 for each deployment, a force multiplier of four is recommended, for every

persistently deployed force package.

The case study above highlights a limitation to this research. Many of the results

discussed in this section are based on analyses we accomplished for AFSOC and, thus,

speciﬁc to our application. That said, similar takeaways to those above can be made, for a

given organization, if adequate analysis is done up front. At the least, to successfully apply

this work, an organization must ﬁrst determine which of the above deﬁnitions is appropriate

and study the organization of interest to determine typical deployment lengths and

availability of personnel.

5. Conclusion

This work derives closed-form equations for determining the force multiplier required to

maintain a speciﬁed deploy-to-dwell ratio (i.e. a quality-of-life metric). The equations relate

the amount of time spent deployed versus at home station, rotation length, transition time

and the total force size, making it possible to analyze the relationships among these factors.

Our methodology provides a way to mathematically justify protecting force production

capacity to sustain enduring deployments while maintaining predetermined combat-

readiness and quality-of-life standards.

The relationships among these factors have important implications for force-

structure logistics. Shorter, more frequent force rotations result in greater amounts of

time deployed overall because of the transition time incurred for each rotation.

Increasing the efﬁciency of force movements, by making transition times shorter,

decreases the overall force size necessary to sustain an enduring deployed capability.

Incomplete force packages and unexpected unavailability lead to uneven

distributions of D2D ratios within units. Therefore, to improve the equitable

distribution of D2D ratios in units, commanders should consider a full force package

committed if any part of it is used to meet demand. Objectively calculating the force

structure necessary to sustain a deployed capability establishes a deﬁnitive supply

cutoff, at which the decision maker risks damaging the production capacity of the

capability. Additionally, using the simple equations derived above, decision-makers

can readily assess force structure logistics decisions in terms of the effects they will

have on units’D2D ratios.

Another signiﬁcant contribution of this work is the development of a standardized

method for calculating the D2D metric. While there are current policies governing the

management of health of the force via D2D, different interpretations can result in different

calculation methods and thus inaccurate comparisons. Standardizing the calculation

facilitates objective equitable comparisons and more effective decision-making.

Furthermore, using the D2D metric for determining deployed capability sustainment

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17

connects force structure decisions with the implications for service members’combat

readiness and quality of life, thus associating the service members’cost with the quantity of

capability demanded.

Although this research focuses on one aspect of readiness, for a single military

organization, over a limited time period, future work could apply the methodology to

other military organizations and evaluate the effectiveness of the implementation of

this research over time. The method could also be extended to determine the force

multiplier based on different readiness or quality-of-life metrics. Additionally, one

could analyze the effect of randomly occurring schedule changes on the distribution of

D2D throughout the units. Finally, creating a simulation to evaluate the long-term

effects of differing types and levels of non-availability on the system, as a whole,

provides another opportunity for future work.

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Corresponding author

Sarah E. Evans can be contacted at: sarah.evans.7@us.af.mil

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