The document discusses how reliability and spare restock time impact the number of spare parts needed to support a system. It uses a notional RF transmitter to examine how doubling the mean time between failures (MTBF) can reduce the required number of spares by half. It also shows that halving the restock time from 2 to 1 month similarly reduces spare requirements. Improved reliability and faster restock times lower costs by reducing the number of spares, repair expenses, storage space needs, and labor. The document demonstrates these concepts using the Spares Poisson Distribution Formula and nomograph.

1. Reliability, Spares, and Other Considerations
Ed Welch
Del Rey Systems & Technology, Inc.
Abstract
Reliability and spare restock time play major roles in determining the number of spares that
should be carried to ensure a spare’s availability. The Spares Poisson Distribution Formula, as
implemented in the 1964 spares nomograph, is an excellent tool to show the effects of reliability
and spare restock time on the recommended quantity of spares for a given spare availability.
A notional rack mounted RF transmitter is under consideration for this study. The variables are
reliability and restock time. Costs (original price, shipping, and repair) in addition to some other
considerations are examined. The study shows that improving reliability reduces the number of
spares required to support a system, spare costs, spare repair costs, storage space, and man-hours.
It is also shown that doubling the item MTBF reduces the number of required on-hand spares by
half. Also, the study shows that reducing the time to restock shows similar reductions.
Acronyms and Notation
CONUS Continental United States
OCONUS Outside Continental United States
MTBF Mean Time Between Failures
λ part failure rate
K number of parts of a particular type
t time
R reliability
1. Introduction
When a system is deployed a common question is: “Do I have enough spares to support my
system?” Usually followed by the statement, “I don’t want too many spares.” Thus we ask, how
can one achieve the right balance?
Reliability and spare restock time play major roles in determining the number of spares that are
carried. Another major consideration in determining on-hand spare quantities is the cost of a
spare (e.g., original price, repair cost, shipping costs, storage costs, and disposal costs).
Reliability and the number of spares required to support equipment failures impact system
availability, the support footprint (e.g., physical space, storage, administrative, damage, disposal,
etc.), and the system life cycle support cost; talk about the proverbial logistics iceberg.
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2. A study of reliability and spare restock time helps to show not only their impact, but also
provides insight into other areas of possible improvement.
In this study we use the Spares Poisson Distribution Formula to calculate the number of spares
needed for a particular application. The formula gives the probability, based on the item
reliability, of having a spare for a particular item available when it is needed:
s  R (− ln R )n 
P = ∑  (1)
n =0  n! 
In (1) S is the number of spare parts carried in stock and K is the number of parts used of a
particular type. The composite reliability R is calculated as,
R = exp(-Kλt) (2)
where λ is the part failure rate; and t is time period of interest. Substituting (2) into (3) yields the
Spares Poisson Distribution formula in terms of the part failure rate and number of parts of a
particular type, and the time period of interest:
s  (Kλt )n e − Kλt 
P = ∑  (3)
n =0  n! 
In 1964 a nomograph was created of the Spares Poisson Distribution Formula and published in
NAVSHIPS 94324, Maintainability Design Handbook for Designers of Shipboard Electronic
Equipment, [1]. The nomograph and its use is also found in Benjamin S. Blanchard’s Logistics
Engineering and Management [2] and Douglas K. Orsburn’s, Spares Management Handbook
[3]. See sidebar.
2. The Effect of Reliability and Restock Time on the Number of Spares
To illustrate the use of the Spares Formula (2) in understanding the effect of the part reliability
on the spares logistics we consider a rack mounted RF transmitter. The RF transmitter is
mounted into a 19 inch rack; the RF transmitter has the following dimensions: 19 inches wide, 5
inches high, and 30 inches in depth; weighs 150 pounds; and costs $40,000 per unit.
The study inputs are:
• annual operation time: 8,760 hours;
• Mean Time Between Failure in hours: 5,000; 10,000, and 20,000;
• number of systems being supported from central storage: 1, 5, 10, 15, 20;
• restock time in months: 1, 2, 3, 4, 5, 6;
• required probability of having a spare when needed: 95%;
• all spare quantities are rounded to the next whole spare number.
Tables 1 – 3 show the effect on the number of spares of varying the reliability, as measured by
MTBF, and restock time for various numbers of systems. The numbers of spares were
determined by using the Spares Poisson Distribution nomograph.
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3. The study shows that improving the reliability (increasing MTBF) reduces the number of
required spares needed to support a system. Restock time is shown as a major spare requirement
driver. For example, by doubling the MTBF the number of required spares is reduced by
approximately half.
Table 1. Numbers of On-Hand Spares to Support Failed Components with MTBF of 5,000
Hours
Number of
Systems 1 5 10 15 20
Restock Months
1 1 2 4 5 6
2 1 4 6 8 9
3 2 5 8 11 13
4 2 6 10 13 17
5 2 7 12 16 20
6 2 8 13 20 24
Table 2. Numbers of On-Hand Spares to Support Failed Components with MTBF of
10,000 Hours
Number of
Systems 1 5 10 15 20
Restock Months
1 1 1 2 3 4
2 1 2 4 5 6
3 1 3 5 6 8
4 2 4 6 8 9
5 2 4 7 9 12
6 2 5 8 11 13
Table 3. Numbers of On-Hand Spares to Support Failed Components with MTBF of
20,000 Hours
Number of
Systems 1 5 10 15 20
Restock Months
1 1 1 1 2 2
2 1 1 2 3 4
3 1 2 3 4 5
4 1 2 4 5 6
5 1 3 4 6 7
6 1 3 5 6 8
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4. 3. The Effect of the Numbers of Spares on Cost
The assumption is that the notional RF Transmitter in this study is repairable. Repair will take
place at the Original Equipment Manufacturer; shipping is within the Continental United States
(CONUS). Cost of repair is assumed to be one half the spare price of $40,000. Thus the total
repair cost for this study for a single spare is $21,800 ($20,000 for repair and $1,800 for
shipping). Figure 2 shows costs based upon the number of on-hand spares (procurement and
approximate annual repair costs). Annual repair costs for RF Transmitters that have 8,760 hours
MTBF or greater would be less. An RF Transmitter with 20,000 hours MTBF would be $10,900
or less.
CONUS shipment time is four days and costs about $900 per shipment (CONUS shipment is the
constant being used in this study); shipment time and cost outside of the Continental United
States (OCONUS) is considerably longer and more costly; additionally, there are various lengths
of time when shipment is not available (e.g., ships at sea and Antarctica).
Figure 2 shows the on-hand spares procurement cost (Inventory) and an approximation for
annual repair costs (Repair). Required spare quantities are found in Tables 1 through 3;
Reliability and restock time cost impact is seen in Figure 2. The number of spares can be
reduced through greater reliability and faster restock time; doubling RF Transmitter reliability
from 10,000 hours MTBF to 20,000 hours MTBF halves the need for spares.
Example: Referring again to Tables 2 and 3(10,000 hours MTBF) and Table 3 (20,000 hours
MTBF), if 10 systems are being supported from a common warehouse and restock time is 2
months the number of required spares decreases from 4 with 10,000 hours MTBF (Table 2) to 2
with 20,000 hours MTBF (Table 3), the cost of spares thus decreases by half (a saving of
$80,000 inventory cost and of $43,600 annual repair costs), storage space decreases by half (a
savings of 13.22 Cubic Feet), and man-hours decreases.
RF Transmitter Spares
$1,200,000
$1,000,000
$800,000
Repair
Cost
$600,000
Inventory
$400,000
$200,000
$0
1 3 5 7 9 11 13 15 17 19 21 23
Quantity
Figure 2 RF Transmitter Spares Costs
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5. 4. The Effect of the Number of Spares on Storage Space
Another major consideration is the space that spares occupy. If the RF Transmitter spares are
kept in a bonded storage room (due to their high value) they occupy a significant amount of
space. The RF Transmitter dimensions are 19 inches wide by 5 inches high by 30 inches in
depth and the unit weighs 150 pounds. Because the RF Transmitters are more fragile (primarily
due to their traveling wave tube construction) they should be shipped in a protective case using
the best commercial practices. A suitable protective case with shock absorbent foam would be
34 inches by 24 inches by 14 inches and weigh about 60 Pounds. Thus, an RF Transmitter spare
would occupy 11,424 cubic inches (6.61 cubic feet) and weight 210 pounds in its protective case.
Figure 3 shows the effect of the number of spares (in protective cases) required storage space.
Doubling the MTBF decreases the spares storage requirement.
Example: Referring to Tables 2 and 3; if 10 systems are being supported from a common
warehouse and the restock time is 2 months the number of spares required can decrease from 4
(Table 2) to 2 (Table 3) and the storage space for spares decreases by half (a savings of 13.22
Cubic Feet).
RF Transmitter Storage
180
160
140
120
Cubic Feet
100
Cubic FT
80
60
40
20
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Quantity
Figure 3 RF Transmitter Spares Storage (In Protective Case)
5. Additional Considerations
For critical communication (commercial or military) systems, a spare should be located onsite in
order to ensure that communication is restored as soon as possible; a switchable parallel system
with separate power sources can be used to ensure there is no loss of communication.
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6. One technique for dealing with the space storage issue is to mount the transmitters into a
standard 19 inch electronic rack that can take the weight; several racks will be required.
An additional consideration is the number of people required to replace the failed transmitter and
to move a transmitter in its protective case. At least two and possibly three people are required
to safely replace a failed transmitter. Four people should be used to move a transmitter when it
is in its case. Ideally, a forklift should be used to reduce the number of required people.
Reducing the number of required spares also reduces the man-hour requirements.
However, it should be noted that spare restock time is not always reducible to a month or less
(e.g., systems located in Antarctica or other remote locations). Also, the cost of improving
reliability can cause the price of having the needed number of spares on hand to become
prohibitive.
6. Conclusions
Reliability has wide reaching cost and logistics impact. In general, greater reliability and faster
restock time reduces the number of spares required to reach a given level of availability (i.e., the
probability that a spare will be available when needed). For example, in our study doubling the
RF Transmitter MTBF from 10,000 hours to 20,000 hours halves the number of spares needed.
This in turn reduces the necessary inventory space and related costs.
Acknowledgment
A condensed version of this paper was published earlier in the RMS Newsletter, October 2007.
References
1. Maintainability Design Handbook for Designers of Shipboard Electronic Equipment,
NAVSHIPS 94324, Washington, D.C., U.S. Department of the Navy, Naval Ship Systems
Command, 1964.
2. Benjamin S. Blanchard, Logistics Engineering and Management, 6th ed. Prentice-Hall, Inc.
2003, 5th ed. 1998.
3. Douglas K. Orsburn, Spares Management Handbook, McGraw-Hill, 1991.
Biography
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7. Ed Welch is a Senior Logistics Analyst at Del Rey Systems & Technology, Inc. and is a part
time Instructor in Logistics at UC San Diego Extension. Ed has more than 30 years of
experience in electrical / electronic systems engineering, logistics engineering, configuration
management, facility planning, reliability and maintainability, technical writing, and supervisory
responsibilities. Ed’s curriculum development experience includes Junior College (logistics and
electronics programs) for San Diego Community College and University (logistic courses) for
University California San Diego.
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8. SIDEBAR
The Spares Poisson Distribution Nomograph
Figure 1. Spare Parts Nomograph. Source: Maintainability Design Handbook for
Designers of Shipboard Electronic Equipment, NAVSHIPS 94324, Washington, D.C., U.S.
Department of the Navy, Naval Ship Systems Command, 1964 [1].
Although expressions such as the Spares Poisson Distribution Formula,
s  (Kλt )n e − Kλt 
P = ∑ 
n =0  n! 
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9. are easily calculated with modern electronic equipment, nomographs are still of much practical
value. If two of the model parameters are known, the third is easily found by drawing a line,
using a straight edge, or stretching a string across two axes of the nomograph and reading the
result from the third axis. The calculation can be done quickly and the visual representation of
the information often gives insights into the effects of varying different parameters that are
difficult to obtain from just the calculated, numeric results.
The Spare Parts Nomograph in Figure 1 contains three graduated lines: “Probability”, P, is the
probability of a spare being available when needed; “Number of Spares”, S, is the number of
required spares; and ΚλΤ is the product of: Κ, the number of parts of a particular type, λ the
failure rate of the part, and T, the time period of interest. To illustrate the use of the nomograph
consider the following example [1]. Let:
K = 20 parts;
λ = 0.1 failure/1000 hours
T = 3 months
Then
KλT = (20)(0.0001)(3)(24)(30) = 4.32
Then for P = 0.95, a straight line from KλT = 4.32 to P = 0.95 crosses the number of spares, S,
line at S = 8 (rounding up). Hence 8 spares are needed to achieve the desired result. As another
example, KλT = 2 and 5 spares gives a 0.98 probability of a spare being available when needed;
increasing S to 6 spares increases the probability of having a spare to approximately 0.994 while
decreasing S to 4 spares reduces the probability to approximately 0.95.
END SIDEBAR
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