Performance Approximation and Design of Pick-and-Pass Order Picking Systems

Abstract

In this paper, we discuss an approximation method based on G/G/m queuing network modeling using Whitt’s (1983) queuing network analyzer to analyze pick-and-pass order picking systems. The objective of this approximation method is to provide an instrument for obtaining rapid performance estimates (such as order lead time and station utilization) of the order picking system. The pick-and-pass system is decomposed into conveyor pieces and pick stations. Conveyor pieces have a constant processing time, whereas the service times at a pick station depend on the number of order lines in the order to be picked at the station, the storage policy at the station, and the working methods. Our approximation method appears to be sufficiently accurate for practical purposes. It can be used to rapidly evaluate the effects of the storage methods in pick stations, the number of order pickers at stations, the size of pick stations, the arrival process of customer orders, and the impact of batching and splitting orders on system performance.

Full text

Performance Approximation and Design of Pick-and-Pass
Order Picking Systems
Mengfei Yu and René de Koster
ERIM REPORT SERIES RESEARCH IN MANAGEMENT
ERIM Report Series reference number ERS-2007-082-LIS
Publication December 2007
Number of pages 32
Persistent paper URL http://hdl.handle.net/1765/10733
Email address corresponding author myu@rsm.nl
Address Erasmus Research Institute of Management (ERIM)
RSM Erasmus University / Erasmus School of Economics
Erasmus Universiteit Rotterdam
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ERASMUS RESEARCH INSTITUTE OF MANAGEMENT
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RESEARCH IN MANAGEMENT
ABSTRACT AND KEYWORDS
Abstract
In this paper, we discuss an approximation method based on G/G/m queuing network modeling
using Whitt’s (1983) queuing network analyzer to analyze pick-and-pass order picking systems.
The objective of this approximation method is to provide an instrument for obtaining rapid
performance estimates (such as order lead time and station utilization) of the order picking
system. The pick-and-pass system is decomposed into conveyor pieces and pick stations.
Conveyor pieces have a constant processing time, whereas the service times at a pick station
depend on the number of order lines in the order to be picked at the station, the storage policy at
the station, and the working methods. Our approximation method appears to be sufficiently
accurate for practical purposes. It can be used to rapidly evaluate the effects of the storage
methods in pick stations, the number of order pickers at stations, the size of pick stations, the
arrival process of customer orders, and the impact of batching and splitting orders on system
performance.
Free Keywords
pick-and-pass, order picking, warehousing, queuing network, simulation
Availability
The ERIM Report Series is distributed through the following platforms:
Academic Repository at Erasmus University (DEAR),
DEAR ERIM Series Portal
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Research Papers in Economics (REPEC), REPEC ERIM Series Webpage
Classifications
The electronic versions of the papers in the ERIM report Series contain bibliographic metadata
by the following classification systems:
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LCC Webpage
Journal of Economic Literature, (JEL), JEL Webpage
ACM Computing Classification System CCS Webpage
Inspec Classification scheme (ICS), ICS Webpage

Performance Approximation and Design of Pick-and-Pass Order
Picking Systems
Mengfei Yu , René de Koster
RSM Erasmus University Rotterdam, The Netherlands
Abstract
In this paper, we discuss an approximation method based on G/G/m queuing network modeling
using Whitt’s (1983) queuing network analyzer to analyze pick-and-pass order picking systems.
The objective of this approximation method is to provide an instrument for obtaining rapid
performance estimates (such as order lead time and station utilization) of the order picking
system. The pick-and-pass system is decomposed into conveyor pieces and pick stations.
Conveyor pieces have a constant processing time, whereas the service times at a pick station
depend on the number of order lines in the order to be picked at the station, the storage policy at
the station, and the working methods. Our approximation method appears to be sufficiently
accurate for practical purposes. It can be used to rapidly evaluate the effects of the storage
methods in pick stations, the number of order pickers at stations, the size of pick stations, the
arrival process of customer orders, and the impact of batching and splitting orders on system
performance.
Key words: Pick-and-pass, order picking, warehousing, queuing network, simulation.
1. Introduction
Order picking, the process of picking products to fill customer orders, is one of the most
important activities in warehouses due to its high contribution (about 55%) to the total
warehouse operating cost (Tompkins et al., 2003). This paper considers a common type of pick-
and-pass order picking system, which consists of a conveyor connecting all pick stations located
along the conveyor line, as sketched in Figure 1. Storage shelves are used to store products at
each pick station. A customer order contains several order lines (an order line is a number of
units of one article). A bin is assigned to a customer order together with a pick list when it
arrives at the order picking system. To fill an order, the order bin is transported on the conveyor
passing various pick stations. If an order line has to be picked at a station, the transportation

system automatically diverts the bin to the station, so that the main flow of bins cannot become
blocked by bins waiting for picking. After entering the pick station, the order bin moves to the
pick position. Order pickers are assigned to pick stations to fill customer orders. An order bin is
processed by one order picker at a station and an order picker works on one order at a time. This
paper assumes the order picker picks one order line per picking trip. The picker starts his trip
from the pick position, reads the next article on the bin’s pick list, walks to the storage shelves
indicated, picks the required article, goes back to the pick position and deposits the picked article
into the bin. Although in some systems multiple lines may be picked in a picking tour, we model
the case where only one article is picked per trip. Systems that we have observed that adhere to
this constraint include a parts Distribution Center (DC) of an international motor production
company (we use this example in our model validation in section 4) where one article is picked
per trip since articles are relatively heavy and need to be barcode scanned. In another warehouse
we studied, even light articles were not batched to reduce pick errors. Having finished the pick
list, the order picker pushes the bin back onto the main conveyor, which transports the bin to a
next pick station. Such pick-and-pass systems are typically applicable in case of a large daily
number of multi-line orders. De Koster (1996) summarizes the advantages of such order picking
systems.
Recent trends in warehouses show that companies tend to accept late orders while providing
rapid and timely delivery within tight time window, which implies time available for order
picking becomes shorter (De Koster et al. 2007). Hence, minimizing order throughput time is an
important objective in many warehouses, and it is used commonly in order picking literature (see
Le-Duc and De Koster 2007, Chew and Tang 1999, and Roodbergen 2001). Exact analysis of a
pick-and-pass system described above is difficult due to the large state space in modeling bin
positions on the conveyor and difficulties in obtaining the exact distribution of service time at
stations. This paper proposes an approximation-based modeling and analysis method to evaluate
the mean order throughput time in such systems. The method provides a fast tool to evaluate
alternatives in designing pick-and-pass systems. Our model relaxes the Jackson queuing network
modeling of De Koster (1994) by allowing a general order arrival process and general service
time distributions, which represent real-life warehouses more accurately and provide a deeper
understanding of the pick-and-pass order picking system. The modeling and the analysis of the
system is based on the analysis of a G/G/m queuing network by Whitt (1983). We show the

approximation method leads to acceptable results by comparing it with both simulation and with
the real order picking process at a parts DC of an international motor production company.
The paper is organized as follows: In section 2, we review literature on order picking, storage,
zoning, and order batching issues. Section 3 describes the approximation model followed by
model validation in section 4. In section 5, we analyze the impact of different warehousing
activities on the system performance. We draw conclusions and discuss possible extensions of
this paper in section 6.
2. Literature review
Literature on order picking processes distinguishes between parts-to-picker and picker-to-parts
picking systems according to whether parts are automatically retrieved by machines and brought
to pick stations for manual picking, or pickers travel along the picking locations to retrieve the
items. A comprehensive literature overview on parts-to-picker order picking systems is given by
Van den Berg (1999), Roodbergen (2001), Le Duc (2005), and Gu et al. (2007). Picker-to-parts
order picking systems are widely used in warehouses. Researchers pay attention to the following
four issues influencing the order picking system performance: storage, batching, routing and
zoning. A recent literature review is given by De Koster et al. (2007).
Storage assignment is the way to assign products to their locations. Mainly three storage policies
are mentioned in literature: random storage, full turn-over based storage, and picking-frequency
class-based storage. In random storage, products are randomly assigned to available storage
locations. Random storage is the simplest way to assign products to their locations and is often
used as a benchmark to compare with other storage policies. In a full-turnover based storage
policy, storage space is reserved for each product according to its turnover rate. A large part of
the literature on full turnover-based storage policies focuses on Cube-per-Order Index (COI)
based storage. The COI of an item is defined as the ratio of the item’s total required space to the
number of trips required to satisfy its demand per period. Articles with low COI are placed
closest to the picking depot (the start and finish position of a picking route). Jarvis (1991) proves
that the COI based storage policy is optimal in minimizing the expected travel distance per order.
Caron et al. (1998) develop an efficient COI-based product-to-location assignment policy with
the objective to minimize the expected travel distance in a picking tour. In practice, pick-
frequency class-based storage is the most popular storage policy used in warehouses. Products

are classified according to their pick frequencies and are stored in classes. Within each class,
items are randomly stored. Petersen et al. (2004) analyze the relation between the number of
product classes and the pickers’ travel time in picker-to-parts order picking systems by
simulation.
Order batching is the process of grouping customer orders together and jointly releasing them
for picking. Order batching reduces the average travel time per order since picking tours are
shared between orders. Gademann et al. (2001) and Gademann and Van De Velde (2005)
consider the order batching problem with objectives to minimize the maximum lead time of a
batch and the total order picking travel time. Both problems are NP-hard and they design
algorithms to solve problems of modest size to optimality. Elsayed and Stern (1983), Hwang et
al. (1988), Gibson and Sharp (1991), Pan and Liu (1995), and Elsayed and Unal (1989) propose
so called seed and saving heuristic algorithms to batch orders to minimize order picking time. De
Koster et al. (1999) perform a comparative study for these algorithms and conclude that even
simple order batching methods lead to significant picking time savings compared to the first-
come first-serve batching rule. Chew and Tang (1999) and Le-Duc and De Koster (2006) set up
stochastic models and use queuing theory to analyze the order batching methods. They provide
bounds and an approximation solution for the average order throughput time.
Routing is the problem to decide the travel route for pickers to retrieve products. It is a special
case of the well-known Traveling Salesman Problem. Ratliff and Rosenthal (1983) use dynamic
programming to find an optimal route for a rectangular, narrow aisles and single-block
warehouse. De Koster et al. (1998) and Roodbergen and De Koster (2001) extend the method for
a warehouse where the I/O point location is decentralized and warehouses with a middle aisle (2
blocks). Instead of the optimal routing methods, heuristics are commonly used in practice
because they are easy to implement and maintain. The two popular heuristic routing methods are
the S-shape (any aisle containing at least one pick is traversed entirely, except potentially the last
visited aisle) and the return heuristics, in which an order picker enters and leaves an aisle from
the same end.
Zoning is the problem of dividing the whole picking area into a number of small areas (zones),
each with one or a few order pickers. The major advantages of zoning are: reduction of the travel
time (because of the smaller traversed area and also the familiarity of the pickers with the zone)
and of the traffic congestion. The analysis on zoning is classified into synchronized zoning,

where all zone pickers work on the same batch of orders at the same time, and progressive
zoning, where each batch of orders (or one order) is processed at one zone at a time. In
progressive zoning, the batch of orders is passed from one zone to the next, which is why such
systems are also called pick-and-pass systems. Jane and Laih (2005) consider heuristics to assign
products to zones with the objective to balance the work load between zones in a synchronized
order picking system. Le-Duc and De Koster (2005) consider the problem of determining the
optimal number of zones (for a given picking area) in a pick-and-pack order picking system to
minimize the mean order throughput time. For progressive zoning, Jane (2000) proposes several
heuristic algorithms to balance the work load among order pickers in zones. De Koster (1994)
approximates pick-and-pass order picking systems by means of Jackson network modeling and
analysis. His model assumes the service time at each pick station is exponentially distributed and
customer orders arrive according to a Poisson process. Jewkes et al. (2004) is the only other
paper considering pick-and-pass order picking system we found. They determine the optimal
pick position of an order bin in a pick station, the optimal product location in pick stations, and
the size of pick stations with the objective of minimizing the order throughput time. Since they
consider a static setting, only travel time to pick orders is considered in their paper. This paper
considers a dynamic setting, where the waiting times of an order bin in front of pick stations is
taken into account.
3. Approximation model
The pick-and-pass order picking system is represented by a sequence of pick stations connected
by conveyor pieces (see Error! Reference source not found.).
Conveyor piece 1
Piece 2
Piece 3
Picking station s
Picking station 1
Picking station 2
Picking station 3
Order bin entrance
Piece 4
Piece s+1
Buffers
Figure 1: Illustration of the pick-and-pass order picking system.
The service time for an order bin at a pick station consists of several components: setup time
(time for starting and finishing the pick list, checking, weighing, labeling, etc.), travel time, and
the picking time for order lines. Travel time depends on the number of order lines to be picked at

the station, the location of these order lines in the pick station, and the travel speed of pickers.
Picking time is proportional to the number of order lines to be picked in the station. We assume
setup time and pickers’ travel speed are constants. We also assume the picking time per order
line, which may consist of multiple units, is constant, and independent of the product type and
the number of units picked. These assumptions will be reasonable when the variance of the units
picked per order line and the pick time itself are small. We suppose a pick-frequency class-based
storage policy (see section 2) in each station. Similar to other research (see e.g. Petersen et al.
2004), we assume demand is uniformly distributed over the products within a product class. The
service time at a pick station is modeled as having a general distribution and is characterized
only by its mean and Squared Coefficient of Variation (SCV). It is reasonable to use only two
moments because in reality service time is hard to fit a theoretical distribution, whereas the
information on mean and the variance of service time is relatively easy to obtain.
A conveyor piece
j
can contain order bins and is assumed to have constant speed, . We
approximate it as servers in parallel, each of which has constant service rate of . This
means that the output rate of a conveyor piece
j
k
j
vl
j
k
jj
kvl /
j
equals exactly if and only if it is completely
full with bins. In the approximation, the output rate of a conveyor piece is proportional to the
number of bins on it. At the end of a conveyor piece, a transition is made by the order bin to the
subsequent conveyor piece, or it is pushed into a pick station. The transition probability of an
order bin to enter a pick station depends on the bin’s pick list and the storage assignment of
products in that station. We approximate this behavior by Markovian transition probability,
which is justified in case of a large number of bins (the typical application area of these systems).
The transition probabilities at the end of a conveyor piece and at leaving a pick station are
calculated in section 3.2. After finishing the picking at a station, the bin is pushed onto a
conveyor piece downstream the pick station.
j
vl
We assume each pick station has infinite storage capacity (buffer) for order bins. This
assumption is reasonable because in reality order pickers at pick stations will ensure that the
system will not be blocked when their stations become full. If a pick station tends to become full,
the order pickers can temporarily put the bins on the floor. We also assume there is a buffer with
infinite capacity in front of each conveyor piece, which means that the arrivals will not be lost
and pick stations and conveyor pieces can not become blocked because of lack of output capacity.

This assumption is also realistic because the conveyor pieces can normally contain a sufficiently
large number of bins.
The whole pick-and-pass order picking system is modeled approximately as a G/G/m queuing
network consisting of nodes preceded by unlimited waiting space in front of them. Nodes
represent conveyor pieces and nodes
SC +
C,...2,1 SCCC
+
+
+
,...2,1
represent pick stations. The
number of servers at each node equals the capacity of each conveyor piece or the number of
order pickers working in the station.
The data used to analyze the queuing network are as follows:
S
: the number of pick stations, with index
j
.
C
: the number of conveyor pieces, with index
j
.
J
: the total number of nodes, equals to
CS
+
, with index
j
.
I
: the number of product classes stored in the pick stations, with index
i
.
N
: the maximum number of order lines contained in a customer order, with index .
n
n
O : probability that an order contains
n
order lines,
Nn ,...2,1
=
.
i
f : order frequency of product class
i
, it is the probability that an order line belongs to the
th product class, . i
Ii ,...2,1=
j
vl
: the velocity of conveyor piece
j
, expressed in bins per second
Cj ,...2,1
=
.
j
k
: the capacity of conveyor piece
j
, expressed in bins.
Cj ,...2,1
=
.
j
h
: the number of order pickers at station
j
,
SCCCj
+
+
+
=
,...2,1
.
j
m
: the number of servers at node
j
,
jj
km
=
for
Cj ,...2,1
=
, and for
.
jj
hm =
SCCCj +++= ,...2,1
ij
l
: the storage space (in meter) used to store products of the
i
th class on the racks at station
j
, ,
Ii ,...2,1= SCCCj
+
+
+= ,...2,1
.
sc
: setup time per bin at a pick station, expressed in seconds.
tp
: picking time for one order line, expressed in seconds.
01
λ
: external arrival rate of order bins to the system, entering node 1, expressed in bins/second.
2
01
c
: SCV of inter-arrival time of order bins to the system.
The variables are as follows:

j
V
: probability of visiting station
j
for an order bin,
SCCCj +
+
+
=
,...2,1
.
j
τ
: total service time at station
j
if the order bin enters station
j
,
SCCCj
+
++
=
,...2,1
.
j
wk
: total travel time at station
j
if the order bin enters station
j
,
SCCCj +++
=
,...2,1
.
j
pk
: total picking time at station
j
if the order bin enters station
j
,
SCCCj
+
++
=
,...2,1
.
2
sj
c
: SCV of service time at node
j
,
SCj
+
=
,...2,1
.
2
aj
c
: SCV of inter-arrival time to node
j
,
SCj
+
=
,...2,1
.
j
λ
: internal arrival rate of order bins to node
j
,
SCj
+
=
,...2,1
.
kj
q
: transition probability from node to node
k
j
,
SCk
+
=
,...2,1
,
SCj +
=
,...2,1
.
j
vt
: number of visits of an order bin to node
j
(either 0 or 1),
SCj +
=
,...2,1
.
j
W
: waiting time of an order bin in front of node
j
,
SCj
+
=
,...2,1
.
j
T
: sojourn time of an order bin at node
j
,
SCj
+
=
,...2,1
.
In the next two subsections, we will derive expressions for the mean and the SCV of the service
time at each node and then calculate the mean throughput time of an order bin in the system.
3.1. Mean and SCV of service times at pick stations and conveyor pieces
The mean service time at station
j
if the order bin enters station
j
, has three components, setup
time , travel time , and the picking time . The mean service time is calculated by
sc
j
wk
j
pk
CjpkEwkEscE
jjj
>∀++= ][][][
τ
(1)
We next derive the expressions for the last two components in equation (1).
The probability that an order line of class
i
is stored in station
j
depends on the order frequency
of the
i
th class products and the space used to stored the
i
th class products in station
j
. It is
given by
Cji
l
l
fp
S
j
ij
ij
iij
>∀∀=
∑
=
,*
1
(2)
Therefore, the probability that an order line is picked in station
j
is the summation of over
i
.
ij
p

∑
=
>∀=
I
i
ijj
CjpP
1
(3)
So the conditional probability of an order bin to enter station
j
given that there are
n
order lines
in the order equals the probability that there is at least one order line to be picked at station
j
:
nCjPV
n
jjn
∀>∀−−= ,)1(1
(4)
Where is the probability that none of the order lines in this order bin is to be picked in
station
n
j
P )1( −
j
. The probability of an order bin to enter station
j
now becomes:
CjOVV
n
N
n
jnj
>∀=
∑
=
*
1
(5)
The number of order lines to be picked in station
j
given that the order bin contains
n
order
lines is a random variable with binomial distribution, i.e.,
CjPP
x
n
orderaninlinesordernxXP
jj
xn
j
x
j
j
jj
>∀−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
==
−
)1(}|{
(6)
Canceling out the condition, we have
CjOPP
x
n
OorderaninlinesordernxXPxXP
n
xn
j
x
j
N
n
j
N
n
njjjj
jj
>∀−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
===
−
=
=
∑
∑
)1(
*}|{}{
1
1
(7)
The expected number of lines to be picked at station
j
given the bin enters station
j
is:
Cj
XP
OPP
x
n
x
XP
xxXP
x
xxXPxxXE
j
N
x
n
xn
j
x
j
N
n
j
j
N
x
j
jjj
j
jjj
N
x
jjj
j
jj
j
j
>∀
=−
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
>
>=
=
>==>
∑∑
∑
∑
=
−
=
=
=
}0{1
)1(
}0{
}0,{
*
}0|{*]0|[
11
1
1
(8)
To obtain the expected travel time, , for an order bin, we need the information of the
products’ locations in a pick station. Under the pick-frequency class-based storage policy, the
optimal locations of products and the picker’s home base (pick position of order bins) in a pick
station is illustrated in
Error! Reference source not found. (see Jewkes et al. 2004), where
][
j
wkE

class A refers to the class of those products with the highest demand frequency, class B the
second highest class, and so on.
Picker’s home base
0.5l1j
A
0.5l
2j
d1j
d2j
0.5lIj
A
0.5l1j0.5l2j0.5lIj
BBI I
dIj
Conveyor
Figure 2: Product locations in the storage rack at station
j
.
The expected travel time at station
j
given that the order bin will enter station
j
is:
CjXXdE
ws
wkE
I
i
jijijj
>∀>=
∑
]0|**2[
1
][
(9)
Where is the travel speed of order pickers expressed in meter/second, is the number of
lines of the
i
th class to be picked at station
ws
ij
X
j
, and is the travel distance from the picker’s
home base to the location of the
i
th class of products. equals
ij
d
ij
X
j
ij
j
P
p
X *
in distribution. As
mentioned before, we suppose that within each class, products are stored randomly and the
demands are uniformly distributed over products. Hence are uniformly distributed random
variables with probability density function of:
ij
d
⎪
⎩
⎪
⎨
⎧
>∀∀≤≤
=
∑∑
=
−
=
elsewhere
Cjilxlfor
l
xf
i
k
kj
i
k
kj
ij
d
ij
0
,
2
1
2
12
)(
0
1
0
(10)
We define in the equation above. Because are independent from , and
0
0
=
j
l
ij
d
j
X
j
ij
P
p
are not
random variables, we obtain
CjdE
P
p
XXE
ws
wkE
I
i
ij
j
ij
jjj
>∀>=
∑
][*]0|[
2
][
(11)
Where, is the expected value of given by
][
ij
dE
ij
d
CjilldxxfxdE
ij
i
k
kjdij
ij
>∀∀+==
∑
∫
−
=
∞
∞−
,
4
1
2
1
)(][
1
0
(12)

Using equation (8), we can calculate the expected picking time at station
j
given that the order
bin will enter station
j
:
CjXXEtppkE
jjj
>∀>= ]0|[*][
(13)
From equation (1), (11) and (13), we can obtain the expected service time at station
j
given that
the order bin will enter station
j
.
To obtain the SCV of service time of an order bin at station
j
, we need to calculate the second
moment of service time, which is given by
222
22
][*2][*2][2][][
])[(][
scpkEscwkEscpkwkEpkEwkE
CjscpkwkEE
jjjjjj
jjj
+++++=
>∀++=
τ
(14)
The second moment of is calculated as follows:
j
wk
CjDEDEDEXXE
ws
DDDEXXE
ws
DEXXE
ws
wkE
I
i
I
ik
I
i
ijkjijjj
I
i
I
ik
I
i
ijkjijjj
I
i
ijjjj
>∀+>=
+>=
>=
∑∑ ∑
∑∑ ∑
∑
=+==
=+==
=
}][][][*2{*]0|[
4
])(*2[*]0|[
4
])[(*]0|[
4
][
11 1
22
2
11 1
22
2
1
22
2
2
(15)
Where
ij
j
ij
ij
d
P
p
D *=
, and
][][
ij
j
ij
ij
dE
P
p
DE =
The last step of equation (15) follows from the independence of and if
ij
D
kj
D
ki
≠
. The
conditional second moment of is given by:
j
X
Cj
XP
OPP
x
n
x
XP
xxXP
xXXE
j
n
xn
j
x
j
N
n
j
N
x
j
j
jjj
N
x
jjj
jj
j
j
>∀
>
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
>
>=
=>
−
==
=
∑∑
∑
}0{
)1(
}0{
}0,{
*]0|[
11
2
1
22
(16)
The second moment of is given below:
ij
D

Cjill
P
p
l
dxx
P
p
l
dE
P
p
DE
i
k
kj
i
k
kj
j
ij
ij
l
l
j
ij
ij
ij
j
ij
ij
i
k
kj
i
k
kj
>∀∀−=
∑
∑
=
=
∑∑
∫
−
==
=
−
=
,])
2
1
()
2
1
[())(
2
(
3
1
*))(
2
(
][*)(][
1
0
3
0
32
2
1
2
1
22
222
0
1
0
(17)
From equation (15) to (17), we obtain . The second moment of is obtained by
][
2
j
wkE
j
pk
]0|[*][
222
>=
jjk
XXEtppkE
(18)
The component is calculated as
][
jj
pkwkE
CjdE
P
p
XXE
ws
tp
XXtpX
P
p
dE
ws
pkwkE
I
i
ij
j
ij
jj
jjj
I
i
j
ij
ijjj
>∀>=
>=
∑
∑
=
=
]}[*{*]0|[*
*2
]0|**)**(*2[
1
][
1
2
1
(19)
From equation (11)-(19), we can obtain the second moment of service time at a pick station
given that the order bin will enter that station. With the value of the first and the second moment
of service time, we can calculate the SCV of service time at station
j
Cj
E
EE
c
j
jj
sj
>∀
−
=
2
22
2
][
][][
τ
ττ
(20)
As mentioned at the beginning of this section, the service rate of each server of a conveyor piece
is constant; therefore the values of SCVs for conveyor pieces are zero, i.e.
Cjc
sj
≤∀= 0
2
(21)
The mean service time of a server on a conveyor piece is the reciprocal of its service rate
Cj
vl
k
E
j
j
j
≤∀=][
τ
(22)
With the information of the mean and the SCV of service time at each node, we will calculate the
order throughput time in the system in the next subsection.

3.2. Mean throughput time of an order
We calculate the mean throughput time of an order bin in the pick-and-pass order picking system
under consideration based on the G/G/m queuing network approximation model of Whitt (1983,
and 1993) (see appendix A). The mean order throughput time consists of transportation times on
conveyor pieces, service times at pick stations, and the waiting times in front of conveyor pieces
and pick stations. The approximation analysis uses two parameters to characterize the arrival
process and the service time at each node, one to describe the rate, and the other to describe the
variability. The two parameters for service time are
][
j
E
τ
, and , as we derived in section 3.1.
For the arrival process, the parameters are
2
sj
c
j
λ
, the arrival rate, which is the reciprocal of the mean
inter-arrival time between two order bins to each node, and , the SCV of the inter-arrival time.
2
aj
c
Orders bins arrive at the system at conveyor piece 1 (see Figure 1) with rate
01
λ
, and the SCV of
the inter-arrival time is . To calculate the internal arrival rate and the SCV of inter-arrival
time at each node, we need to know the transition probabilities between nodes. At the end of a
conveyor piece, an order bin is either transferred to a subsequent conveyor piece for
transportation or pushed into a pick station. The transition probabilities between these nodes are
given by
2
01
c
CjVq
CjCjj
<∀=
++
(23)
CjVq
Cjjj
<∀−=
++
1
1
(24)
SCjCq
Sjj
+≤<∀=
−
1
(25)
Where the value of is obtained from equation
Cj
V
+
(5). The transition probabilities between other
nodes are zero. Because order bins leave the system from the last conveyor piece
C
, we have
. The matrix of the transition probabilities is indicated by
Q
. As an
example, consider a network with 3 pick stations and 4 conveyor pieces, i.e., , and
Jjforq
jC
≤≤∀= 10
4=C 3
=
S
.
Assuming that at the end of each conveyor piece (except for piece 4, the last one), a bin has a
probability of 0.6 to be pushed into the next pick station. Bins enter the system from node 1 and
leave the system from node 4. The Markov transition matrix is then given by

⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
=
0001000
0000100
0000010
0000000
6.0004.0000
06.0004.000
006.0004.00
Q
With the probability transition matrix, we can obtain the internal traffic rates
j
λ
and the SCV of
the inter-arrival time between two bins to each node (see Appendix A).
The utilization of a conveyor piece and a pick station is given by
⎪
⎩
⎪
⎨
⎧
>∀
≤∀
=
CjhE
Cjvl
jjj
jj
j
/][
/
τλ
λ
ρ
(26)
The expected sojourn time of a bin at node
j
is given by
JjEWEvtETE
jjjj
≤
≤
∀
+= 1])[][(*][][
τ
(27)
Where is the expected waiting time in front of node
][
j
WE
j
as calculated by (A.9), and
is the expected number of visits to node
][
j
vtE
j
of an order bin. The probability mass function of
is given by
j
vt
Jj
Vyprobabilitwith
Vyprobabilitwith
vt
j
j
j
≤≤∀
⎪
⎩
⎪
⎨
⎧
−
= 1
1
10
(28)
Where is obtained from equation (5) for and
j
V
Cj >
1
=
j
V
for
Cj
≤
. Hence
JjVVVvtE
jjjj
≤
≤
∀=+−= 1)1(*0][
(29)
The total expected order throughput time is the summation of the expected sojourn time at each
node.
4. Model validation
To validate the quality of the approximation method described in section 3, we compare the
results with both simulation and a real order picking process.
We built a simulation model in Automod
®
10.0. For each scenario in the example, we use at least
20,000 orders, preceded by 2000 orders of initialization for the system to become stable, to

guarantee that the 95%-confidence interval width of the Mean Order Throughput Time (MOTT)
is below 1% of the mean value. The parameters used in the example are listed in Table 1.
Table 1: Parameters used in the example.
Parameter Value
Order arrival process Poisson distributed (we evaluate different arrival
rates)
Number of stations 18
Number of order pickers 18
Product classes and order frequency per class Class 1: f
1
=0.8, Class 2: f
2
=0.15, Class 3: f
3
=0.05
Total fraction of storage space for product classes Class 1: 0.2, Class 2: 0.3, Class 3: 0.5
Size of order bins 60*40*35 cm
Conveyor speed 0.7 bins per second (0.1m minimum space
between two bins)
Conveyor length First piece 40 bins, 20 bins for others
Length of each pick station 28 meters (40 bins)
Walk speed of order pickers 1 meter/second
Picking time per line 18 seconds
Setup time 45 seconds
Maximum number of lines in an order bin 30
The number of order lines in an order Empirical distribution (based on the data from a
Dutch warehouse) with mean of 15.6 and standard
deviation of 6.3
Table 2 illustrates the storage assignments in stations and the probability that an order bin has to
be handled at a station. We observe from Table 2 that stations have the same total storage space
but use different storage space per product class (i.e., a non-uniform storage policy).
Table 2: Storage space and the bin visit probabilities to stations under the non-uniform storage
policy.
l
ij
(meter) St. 1 St.2 St. 3 St. 4 St. 5 St.6 St. 7 St. 8 St. 9
clas1 4.9 5.6 6.3 4.9 5.6 6.3 4.9 5.6 4.9
clas2 7.7 8.4 9.1 7.7 8.4 9.1 7.7 8.4 7.7
clas3 15.4 14 12.6 15.4 14 12.6 15.4 14 15.4
Bin visit prob. 0.36 0.39 0.43 0.36 0.39 0.43 0.36 0.39 0.43
St. 10 St.11 St.12 St. 13 St. 14 St.15 St. 16 St. 17 St. 18
clas1 6.3 4.9 5.6 6.3 4.9 5.6 6.3 4.9 5.6
clas2 9.1 7.7 8.4 9.1 7.7 8.4 9.1 7.7 8.4
clas3 12.6 15.4 14 12.6 15.4 14 12.6 15.4 14
Bin visit prob. 0.36 0.39 0.43 0.36 0.39 0.43 0.36 0.39 0.43
We vary the order arrival rates to the system to compare the performance of the approximation
method to simulation under different work loads. The results are listed in Table 3. Table 3 also
illustrates the accuracy of G/G/m modeling over Jackson modeling used in De Koster (1994).

Table 3: Validation results for the example and comparisons to Jackson modeling.
Input rate
(bin/sec)
MOTT (sec)(G/G/m) MOTT(sec)
(Jackson)
Numerical Simulation Rel. error Station
utilization (max)
0.008 1615.5 1556.2±4.6 3.81% 0.409 1867.5
0.011 1725.0 1647.3±5.2 4.72% 0.517 2119.9
0.013 1889.8 1789.5±6.1 5.60% 0.630 2518.6
0.016 2290.8 2171.5±8.3 5.49% 0.780 3559.0
0.018 3116.0 3023.4±15.7 3.06% 0.893 5792.4
0.019 4312.8 4247.4±24.4 1.54% 0.944 9078.5
Table 3 shows that the relative errors between the approximation model and the simulation
results are all below 6 percent under different work loads. It also shows that the larger the
utilizations at stations, the more accurate G/G/m modeling over Jackson modeling.
We also conducted other experiments with different parameters: the number of pick stations
varied from 4 to 18, with a step size of 2, and the utilization of pick stations varied from 0.2 to
0.9 with step size of 0.1. In all experimental settings, the relative error between the
approximation model and the simulation results were below 7 %.
To further validate our approximation method, we compare our results to the performance of a
real order picking process in the bulky storage area at the parts distribution center of an
international motor production company. The bulky storage area stores in total 240 products
divided into 3 classes. One class contains 48 heavy products and the other two classes are
categorized according to their order frequencies, each containing 96 products. The whole area is
divided into four pick stations connected by conveyor pieces. Through analyzing the log files
from the Warehouse Management System (WMS) for a picking day, which is chosen as a
representative of its typical picking process, we obtained the data for the order arrival process to
the system, the service times at pick stations, and the routing probabilities of order bins to enter
each station. The results are listed in Table 4. We also measured the capacities of conveyor
pieces and their moving speeds. We input these data into our approximation model. The result of
MOTT is compared with the mean order throughput time obtained from the warehouse
management system.

Table 4: Data and comparison with results of the real order picking system.
Parameter Value
Number of stations 4
Number of order pickers per station 1
Number of order lines to pick per order Empirical distribution (mean, 2.5 , stdv, 1.9 )
Order inter-arrival time to the system (sec) Empirical distribution (mean, 28.9, stdv, 52.4)
Service time at station A (sec) Empirical distribution (mean, 40.1, stdv, 41.6)
Service time at station B (sec) Empirical distribution (mean, 51.0, stdv, 51.1)
Service time at station C (sec) Empirical distribution (mean, 54.1, stdv, 48.0)
Service time at station D (sec) Empirical distribution (mean, 38.8, stdv, 35.0)
Prob. To enter station A 0.385
Prob. To enter station B 0.254
Prob. To enter station C 0.271
Prob. To enter station D 0.435
MOTT from G/G/m approximation model (sec) 302.1
MOTT from WMS (sec) 321.7
Relative error 6.1%
From Table 4, we find that the relative error is around 6 percent. We conclude that the quality of
the approximation method is acceptable for practical purposes. In the next section, we use this
approximation method as a tool to estimate the pick-and-pass order picking system performance
under the various warehousing policies.
5. Scenario analyses
In this section, we use the approximation method to analyze the impact of different warehousing
policies on the order picking system performance. These policies include the storage assignments
in pick stations, the size of pick stations, the number of order pickers in stations, and order
batching and splitting decisions in the order release process. The parameters used for these
scenario analyses are illustrated in Table 1.
5.1. The effects of storage policies on system performance
Storage policies affect the order throughput time in the pick-and-pass system, as they impact the
work load balance between stations. In this subsection, we will compare the impact of uniform
(stations use identical storage spaces to store a certain class of products) and non-uniform
(stations use different storage spaces to store a certain class of products) storage policies on
mean order throughput time. We expect that the uniform storage policy leads to shorter order
throughput time, as it leads to work load balance between stations.

The storage space for each class of products in stations, and the probability for a bin to enter a
pick station under the uniform storage policy are shown in Table 5.
Table 5: Storage space and the bin visit probabilities to stations under the uniform storage policy.
l
ij
(meter) St. 1 St.2 St. 3 St. 4 St. 5 St.6 St. 7 St. 8 St. 9
clas1 5.6 5.6 5.6 5.6 5.6 5.6 5.6 5.6 5.6
clas2 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4
clas3 14 14 14 14 14 14 14 14 14
Bin visit prob. 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39
St. 10 St.11 St.12 St. 13 St. 14 St.15 St. 16 St. 17 St. 18
clas1 5.6 5.6 5.6 5.6 5.6 5.6 5.6 5.6 5.6
clas2 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4 8.4
clas3 14 14 14 14 14 14 14 14 14
Bin visit prob. 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39
Table 6 illustrates the comparison with the non-uniform storage policy (refer to Table 2 and
Table 3). As the stations are now balanced on average, we find from Table 6 that the mean order
throughput times are shorter under the uniform storage policy than under the non-uniform
storage policy. The improvement is substantial when the work load of the system increases.
Table 6: Comparison of system performance between uniform and non-uniform storage policies
in pick stations.
MOTT(sec) Utilization
Input rate (bin/sec) Uniform Non-uniform Improvement Uniform Non-uniform
0.008 1613.0
1615.5
0.15% 0.376
0.409
0.011 1720.3
1725.0
0.27% 0.475
0.517
0.013 1876.4
1889.8
0.71% 0.579
0.630
0.016 2236.6
2290.8
2.37% 0.716
0.780
0.018 2849.1
3116.0
8.57% 0.821
0.893
0.019 3436.9
4312.8
20.31% 0.868
0.944
Because of the advantage of the uniform storage policy, we will focus our analysis on this
storage policy in the following discussions.
5.2. The effects of station sizes and the number of pickers on system performance
The size of the pick stations and the number of order pickers in stations impact the mean order
throughput time. With a fixed length of the whole order picking system (i.e., a fixed storage
capacity of the system) and a fixed number of order pickers, the larger the size of the pick
stations, the fewer number of stations we have in the system, and the more order pickers are

available at each pick station. Pick stations of larger size will increase the service time due to
longer picking travel time, and the fewer number of stations tends to increase the utilizations of
pick stations due to higher order bin arrival rates. Therefore they lead to an increase of the mean
order throughput time. But on the other hand, fewer number of stations leads to fewer station
visits of an order bin (hence less queues and less setup time); more order pickers per station
implies decreasing utilizations at pick stations, which reduces the mean order throughput time. In
pick-and-pass order picking system design, a main question therefore is to find the right trade-off
between these opposite effects by selecting the right number of stations. Table 7 shows the
system performance for various combinations of station sizes and order pickers per station. It
shows that under the current settings, the scenario of 6 stations with 3 order pickers per station
has the best performance in all possible alternatives.
Table 7:
System performances under various station sizes and the number of order pickers per
station.
# of stations(# of picker per station)(station size in meters)
18(1)(28) 9(2)(56) 6(3)(84)
Input rate
(bin/sec)
MOTT
(sec)
Utilization MOTT
(sec)
Utilization MOTT
(sec)
Utilization
0.008 1613.0 0.376 1370.7 0.348 1330.2 0.345
0.011 1720.3 0.475 1407.9 0.439 1351.7 0.436
0.013 1876.4 0.579 1463.7 0.535 1386.6 0.531
0.016 2236.6 0.716 1586.9 0.663 1468.9 0.657
0.018 2849.1 0.821 1765.7 0.759 1591.7 0.753
0.019 3436.9 0.868 1904.3 0.803 1687.0 0.796
0.020 4226.5 0.903 2052.3 0.835 1788.2 0.828
0.021 6110.2 0.940 2294.8 0.870 1951.6 0.863
# of stations(# of picker per station)(station size in meters)
3(6)(168) 2(9)(252) 1(18)(514)
Input rate
(bin/sec)
MOTT
(sec)
Utilization MOTT
(sec)
Utilization MOTT
(sec)
Utilization
0.008 1426.6 0.398 1607.1 0.482 2304.4 0.771
0.011 1441.2 0.503 1630.3 0.608 4867.1 0.974
0.013 1474.0 0.612 1706.1 0.741 inf >1
0.016 1587.6 0.758 2332.0 0.917 inf >1
0.018 1870.8 0.868 inf >1 inf >1
0.019 2263.9 0.918 inf >1 inf >1
0.020 3113.5 0.955 inf >1 inf >1
0.021 16765.0 0.994 inf >1 inf >1

5.3. The effects of batching orders on system performance
As we have seen from the analysis above, the input rate of order bins to the system has great
impact on system performance. A large arrival rate results in higher work load to the system, and
will subsequently increase the mean order throughput time. One way to reduce the input rate to
the system is to batch orders. We consider the following batching rules: We batch two successive
order bins each containing at most
L
lines into one bin, and then send it to the system. The order
bins with larger than
L
lines are sent directly to the system. The batching threshold,
L
can take
any value between 1 and
⎥
⎦
⎥
⎢
⎣
⎢
2
N
, where
⎣
⎦
* means rounding down to the nearest integer.
Otherwise, the number of lines in a batched bin may exceed the bin’s capacity. We assume that
(the maximum number of lines in an order) is also the capacity of an order bin. By batching
small orders, we can decrease the input rate to the system, leading to decrease the mean order
throughput time. On the other hand, the service time at each station and the probability of
entering a pick station will increase because of more order lines to be picked. These factors lead
to increase the mean order throughput time. When we batch two successive bins with fewer than
L lines, the first bin has to wait for several inter-arrival time periods to be processed. However,
since the mean order inter-arrival time is normally very small compared to the total mean order
throughput time, and only those bins containing less than L lines are batched, this effect is small
and can be neglected. The impact of order batching on system performance depends on the trade-
off between these factors. We can analyze this impact with a slight modification of the
approximation method discussed above.
N
Assuming the original input process to the system is Poisson distributed with rate
01
λ
, an order
bin has a probability to contain
n
order lines. The flow of order bins with
n
order lines is
also a Poisson process with rate
n
O
n
O*
01
λ
. After batching, the original process is split into two
sub-processes. The first sub-process refers to the batched bins, and the second sub-process is the
un-batched bins. According to the properties of Poisson process, the inter-arrival time of the first
sub-process is Gamma distributed with parameters (2, ). The input rate of this type of
order flow is
∑
=
L
n
n
O
1
01
*
λ
∑
=
=
L
n
n
O
1
01
011
~
*
2
1
λλ
, and the SCV of the order inter-arrival time is . The
5.0
2
011
=c

second sub-process is Poisson distributed with rate , where
N
is the
maximum number of lines in a bin. The SCV of the order inter-arrival time is .
∑
+=
=
N
Ln
n
O
1
01
012
~
*
λλ
1
2
012
=c
The basic idea to calculate the mean order throughput time with two input flows is derived from
Whitt (1983). The procedure is first to calculate the mean and the SCV of service time at each
pick station, the transition probabilities between nodes, and the internal traffic flows to each node
separately for each input flow, and then we convert these two types of flows into one (See
Appendix B). The method of Appendix A is again used to obtain the mean order throughput time.
Following the example at the beginning of this section, we assume that
L
equals 15. Table 8
compares the system performance between batching and non-batching scenarios.
Table 8: Comparison of system performances between batching and non-batching scenarios.
L=15
Order arrival rate (bins/sec) 0.0083 0.0105 0.0128 0.0159 0.0182 0.0185
Rate after batching (bins/sec) 0.0063 0.0079 0.0096 0.0119 0.0136 0.0138
Batching 1864.2 1973.9 2123.4 2435.8 2891.8 2971.6
MOTPT (sec)
Non-batching 1613.0 1720.3 1876.4 2236.6 2849.1 2968.2
Batching 0.358 0.452 0.551 0.682 0.781 0.792
Utilization
Non-batching 0.376 0.475 0.579 0.716 0.821 0.833
Batching 22.1 31.0 43.1 68.4 105.3 111.7 Mean waiting time
(sec)
Non-batching 23.7 34.3 49.7 85.2 145.5 157.3
Batching 83.4 83.4 83.4 83.4 83.4 83.4 Mean service time
(sec)
Non-batching 80.1 80.1 80.1 80.1 80.1 80.1
Batching 0.69 0.69 0.69 0.69 0.69 0.69
Bin visiting prob.
Non-batching 0.56 0.56 0.56 0.56 0.56 0.56
L=15
Order arrival rate (bins/sec) 0.0186 0.0189 0.0192 0.0200 0.0204 0.0208
Rate after batching (bins/sec) 0.0139 0.0142 0.0144 0.0150 0.0153 0.0156
Batching 3004.5 3114.5 3162.0 3679.9 3991.1 4424.2
MOTPT (sec)
Non-batching 3018.4 3191.7 3436.9 4226.5 4925.5 6110.2
Batching 0.797 0.810 0.826 0.859 0.876 0.895
Utilization
Non-batching 0.837 0.852 0.868 0.903 0.921 0.940
Batching 114.4 123.3 135.2 169.0 194.2 229.2 Mean waiting time
(sec)
Non-batching 162.2 179.3 203.5 281.3 350.1 466.9
Batching 83.4 83.4 83.4 83.4 83.4 83.4 Mean service time
(sec)
Non-batching 80.1 80.1 80.1 80.1 80.1 80.1
Batching 0.69 0.69 0.69 0.69 0.69 0.69
Bin visiting prob.
Non-batching 0.56 0.56 0.56 0.56 0.56 0.56
Table 8 shows that the input rates decrease, and the service times at pick stations increase when
orders are batched. Batching orders can slightly reduce the utilizations of pick stations. The

impact of pick station utilizations on waiting times in front of stations is marginal when the
utilizations are low, but becomes substantial when the utilizations get higher. We observe that
when the system is not heavily loaded, order batching increases the mean order throughput time.
This is mainly due to the longer service time at pick stations, and the increased probability of
entering pick stations. However, when the system is heavily loaded, the mean order throughput
time decreases when we batch orders. Under a heavy load, waiting time is the major component
of the order throughput time; reducing pick stations utilizations by batching orders can
significantly reduce waiting time in front of pick stations, and therefore reduces the mean order
throughput time.
5.4. The effects of splitting orders on system performance
As an alternative to batching orders, splitting an order into two small orders will reduce the order
bin service times in pick stations and the probabilities of entering pick stations. On the other
hand, splitting orders increases the arrival flow rate because more order bins enter the system. To
analyze the impact of order splitting on system performance, we split an order bin containing
R
or more than
R
lines into two bins, one containing
⎣
⎦
2/R lines and the other containing
R
-
lines. Again, assuming the original arrival process is Poisson distributed, the input
process is divided into two Poisson processes: the input flow of non-split bins with rate
, and the input flow of bins to be split with rate . Before
arriving at the first conveyor piece, we suppose the input flow of bins to be split will first pass
through an artificial node with very small constant service time. A new order bin is created
following the completion of service at the artificial node. According to the approximation
method given at section 2.2 and 4.6 of Whitt 1983, the departure process, i.e., the arrival process
to the first conveyor piece of this flow of split bins has rate of , and approximated SCV
of inter-arrival time of 2.
⎣
2/R
⎦
=
01
011
~
*
R
n
n
λλ
∑
−
=
1
1
O
∑
=
=
N
Rn
n
O*
01
012
~
λλ
012
~
*2
λ
The total arrival process to the first conveyor piece is therefore the combination of a Poisson
process, with rate , and a process with rate of and SCV of inter-arrival
time of 2. Similar to the approaches used to analyze batching orders, we can obtain the system
∑
−
=
=
1
1
01
011
~
*
R
n
n
O
λλ
012
~
*2
λ

performance for the order splitting scenario. The results with comparison to the non-splitting
scenario, as illustrated in Table 9, show that splitting orders increases the input rate to the system
and reduces the service times at pick stations and the probabilities of entering pick stations.
Splitting orders increases the utilizations of pick stations. The mean order throughput time
shortens when the station utilizations are low. This is mainly due to the reduction in service
times and the probabilities of entering pick stations. When station utilization becomes high
(
75.0>
ρ
approximately for R equals 15), order splitting increases the mean order throughput
time because the waiting time in front of a station becomes longer due to higher utilization.
Table 9: Comparison of system performances between splitting and non-splitting scenario.
R=15
Order arrival rate (bins/sec) 0.0083 0.0105 0.0128 0.0159 0.0164 0.0166
Rate after splitting (bins/sec) 0.0130 0.0164 0.0200 0.0248 0.0256 0.0259
Splitting 1344.6 1451.1 1624.1 2128.2 2295.8 2369.6
MOTPT (sec)
Non-splitting 1613.0 1720.3 1876.4 2236.6 2333.2 2372.8
Splitting 0.416 0.526 0.640 0.793 0.819 0.828
Utilization
Non-splitting 0.376 0.475 0.579 0.716 0.740 0.749
Splitting 26.9 40.5 63.0 128.2 149.8 159.3
Mean waiting time
(sec)
Non-splitting 23.7 34.3 49.7 85.2 94.7 98.6
Splitting 74.5 74.5 74.5 74.5 74.5 74.5
Mean service time
(sec)
Non-splitting 80.1 80.1 80.1 80.1 80.1 80.1
Splitting 0.43 0.43 0.43 0.43 0.43 0.43
Bin visiting prob.
Non-splitting 0.56 0.56 0.56 0.56 0.56 0.56
R=15
Order arrival rate (bins/sec) 0.0167 0.0169 0.0172 0.0175 0.0182 0.0192
Rate after splitting (bins/sec) 0.0260 0.0264 0.0269 0.0274 0.0284 0.0300
Splitting 2404.3 2536.4 2700.9 2911.8 3581.9 7023.3
MOTPT (sec)
Non-splitting 2390.9 2456.6 2532.3 2620.4 2849.1 3436.9
Splitting 0.833 0.847 0.861 0.876 0.908 0.961
Utilization
Non-splitting 0.752 0.765 0.778 0.792 0.821 0.868
Splitting 163.8 180.9 202.2 229.4 316.0 760.6
Mean waiting time
(sec)
Non-splitting 100.4 106.8 114.3 123.0 145.5 203.5
Splitting 74.5 74.5 74.5 74.5 74.5 74.5
Mean service time
(sec)
Non-splitting 80.1 80.1 80.1 80.1 80.1 80.1
Splitting 0.43 0.43 0.43 0.43 0.43 0.43
Bin visiting prob.
Non-splitting 0.56 0.56 0.56 0.56 0.56 0.56
We note that the approximation model underestimates the mean order throughput time when we
consider each split as a separate order. However, in reality, orders are only split when the
number of order lines is large, and the impact on mean throughput time will be slight. The

approximation model will give a reasonable estimation for the mean order throughput time from
a practical point of view.
6. Conclusions and extensions
In this paper, we propose an approximation method based on G/G/m queuing network modeling
to analyze performance of a pick-and-pass order picking system. The method can be used as a
fast tool to estimate design alternatives on the mean order throughput time of the order picking
system. These alternatives include the storage policies, the size of pick stations, the number of
order pickers in stations, and the arrival process of customer orders. In general, the preference of
one alternative over others is subject to a detailed specification of the order picking system. The
quality of the approximation method is acceptable for practical purposes. Therefore it enables
planners to evaluate various system alternatives, which is essential at the design phase of the
order picking system. Additionally, the approximation method can also be used to evaluate
various operational policies like order batching and order splitting on system performance.
The model lends itself to several modifications and extensions left for future research. Although
we assumed in this paper pickers pick only one order line in their picking tour, it is possible to
relax this assumption and derive the first and second moment of service time for picking multiple
lines in a pick tour. We may also take the number of units to pick in an order line into
consideration and differentiate the picking time for different articles. In such cases, the number
of units to pick in an order line and the picking time per article are both stochastic variables. The
G/G/m queuing network approximation model still can be used to analyze these situations, but
characterizing the distribution is less straightforward. The G/G/m queuing network
approximation model still can be used to analyze this situation. The layout of pick stations can be
altered (we here assumed a line layout) to, for example, a parallel-aisle layout. We also estimated
the standard deviation of order throughput time using the method described in Whitt (1983).
However, the method did not provide good estimation results. It would also be interesting to find
a more accurate approach to estimate the standard deviation of order throughput time, which
together with mean order throughput time provides a better description of the order picking
system performance. Another interesting extension of the paper is to consider the situation that
an order picker is responsible for the pickings at multiple pick stations. Furthermore, in reality,
the buffer capacity in front of each pick station is finite, which influences performance in high-

utilization situations. It might be possible to derive estimates for the mean throughput time using
approximation methods for finite buffer queuing networks.
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Appendix A
According to Whitt (1983), to estimate the mean order throughput time in this G/G/m queuing
network system, we need to calculate the internal flow parameters. The internal flow rate to each
node,
j
λ
, is obtained by solving the following linear equations
Jjq
ij
J
i
ijj
≤≤+=
∑
=
1
1
0
λλλ
(A. 1)
Where
j0
λ
is the external arrival rate to node
j
, is the total number of nodes (conveyor
pieces and pick stations) in the system, and is the transition probability from node
i
to node
J
ij
q
j
.
The arrival rate to node
j
from node is given by i
JjJiq
ijiij
≤
≤∀≤≤∀= 1,1
λ
λ
(A. 2)
The proportion of arrivals to
j
that come from , is calculated by
i
JjJipr
jijij
≤
≤∀≤≤∀= 1,0/
λ
λ
(A. 3)
The variability parameters of the internal flow, i.e. the SCVs of the inter-arrival time of the
arrival processes to nodes, are calculated by solving the following linear equations
Jjbcac
ij
J
i
aijaj
≤≤+=
∑
=
1
1
22
(A. 4)
where
⎭
⎬
⎫
⎩
⎨
⎧
+−+−+=
∑
=
J
i
iiijijijjjjj
xqqprcpra
1
22
00
]})1[()1(1
ρω
(A. 5)
)1(
2
iijijjij
qprb
ρω
−=
(A. 6)
2
0 j
c
is the SCV of the external inter-arrival time to node
j
, and , since the
order bins enter the system from the first conveyor piece.
10
2
0
>∀= jforc
j
i
ρ
is the utilization of node obtained from equation (26), and i
)1}2.0,(max{1
25.0
−+=
−
siii
cmx
(A. 7)
with the number of servers at node
i
, and the SCV of service time at node
i
obtained
from equation (20) and (21).
i
m
2
si
c

12
)]1()1(41[
−
−−+=
jjj
v
ρω
(A. 8)
with .
1
0
2
][
−
=
∑
=
J
i
ijj
prv
With the internal flow parameters,
j
λ
and , and the service time parameters,
2
aj
c
][
j
E
τ
, and ,
Whitt (1983) decomposes the network into separate service facilities that are analyzed in
isolation. Each service facility is a G/G/m queue. Whitt (1993) provides the following
approximation for the expected waiting time in queues. Since we are focusing on a single node,
we omit the subscript indexing the node in deriving the expected waiting time in front of a node.
2
sj
c
For a multi-server node with servers, the expected waiting time is given by
m
mMM
sa
samGG
WE
cc
mccWE
//
22
22
//
][
2
),,,(][
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
=
ρφ
(A. 9)
where and are obtained from (A.4), and equation (20) and (21) respectively,
2
a
c
2
s
c
ρ
is given
by equation (26), is the waiting time in queue of a multi-server node with Poisson
arrivals and exponential service distribution. The exact expression for is given by
mMM
WE
//
][
mMM
WE
//
][
)1(
)(
][
//
ρμ
−
≥
=
m
mNP
WE
mMM
(A. 10)
where
μ
is the reciprocal of mean service time at each node.
)( mNP ≥
is the probability that all servers are busy and is given by
ζ
ρ
ρ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=≥
)1(!
)(
)(
m
m
mNP
m
(A. 11)
with
1
1
0
!
)(
)1(!
)(
−
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
=
∑
m
k
km
k
m
m
m
ρ
ρ
ρ
ζ
The expression for
φ
in (A.9) is given by
⎪
⎪
⎩
⎪
⎪
⎨
⎧
≤
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
≥
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
=
222
22
22
3
22
22
222
22
2
1
22
22
22
),,(
(2
3
),(
(2
),,(
34
),(
34
)(4
),,,(
sa
sa
as
sa
as
sa
sa
s
sa
sa
sa
ccmc
cc
cc
m
cc
cc
ccmc
cc
c
m
cc
cc
mcc
ρψρφ
ρψρφ
ρφ
(A. 12)
with
),(1),(
1
ρ
γ
ρ
φ
mm +=
(A. 13)

ρ
ρ
ργρφ
3
)1(2
3
)),(41(),(
−−
−= emm
(A. 14)
}
16
]2)54)[(1)(1(
,24.0min{),(
5.0
ρ
ρ
ργ
m
mm
m
−+−−
=
(A. 15)
and
⎪
⎩
⎪
⎨
⎧
≤≤
>
=
−
10),(
11
),,(
2)1(2
4
2
2
2
cm
c
mc
c
ρφ
ρψ
(A. 16)
with
2
22
2
sa
cc
c
+
=
and
}
2
),(),(
,1min{),(
31
4
ρ
φ
ρ
φ
ρφ
mm
m
+
=
.

Appendix B
Based on the work of Whitt (1983), we convert the two input flows into one. The external arrival
rate to the system is given by
012
~
011
~
01
λλλ
+= (B. 1)
where
01
λ
is the combined external arrival rate to the system, and are the two separate
external arrival rates to the system. The internal traffic rate to node
011
~
λ
012
~
λ
j
is given by
Jj
jj
j
≤≤∀+= 1
2
~
1
~
λλλ
(B. 2)
where and are the internal traffic rates to node
1
~
j
λ
2
~
j
λ
j
from each input flow solved by the
linear equations of (A.1).
The mean service time at pick station
j
is the weighted combination of the service times for two
separate input flows
Cj
EE
E
jj
j
j
j
j
j
>∀
+
+
=
2
~
1
~
2
2
~
1
1
~
][][
][
λλ
τλτλ
τ
(B. 3)
where
][
1j
E
τ
and
][
2j
E
τ
are the mean service time for each separate input flow derived from
equation (1).
The second moment of service time at pick station
j
is derived by
Cj
EE
E
jj
j
j
j
j
j
>∀
+
+
=
2
~
1
~
2
2
2
~
2
1
1
~
2
][][
][
λλ
τλτλ
τ
(B. 4)
where and are the second moments of service time at pick station
][
2
1
j
E
τ
][
2
2
j
E
τ
j
for each
input flow given by equation (14).
The SCV of service time at pick station
j
, , can then be calculated from equation (20), (B.3),
and (B.4). Because the service time is constant at conveyor pieces, the SCV and the mean of
service time are obtained from equation (21) and (22).
2
sj
c
The SCV of inter-arrival time to each node, , is again obtained from (A.4). The required
parameters are calculated as follows:
2
aj
c

The transition probabilities from node i to node
j
are calculated as
⎩
⎨
⎧
−=≤≤+∀
+=+=≤≤∀
=
CijJiC
CijandijCi
q
ij
ij
,11
,1,1/
01
λλ
(B. 5)
ij
λ
, the arrival rate from node
i
to node
j
is given by
JjJi
ijij
ij
≤≤∀≤≤∀+= 1,1
2
~
1
~
λλλ
(B. 6)
where and are the arrival rates from node
i
to node
1
~
ij
λ
2
~
ij
λ
j
for each separate input flow
derived from (A.2).
The utilizations
j
ρ
at each node
j
, are calculated from equation (26).
ij
pr
, the proportion of arrivals to
j
that come from ( ), is obtained from (A.3).
i
0≥i
The SCV for the inter-arrival time of orders to the system is given by
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
+
+
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
+
+−=
012
~
011
~
012
~
2
012
012
~
011
~
011
~
2
011
1
~
1
~
2
01
)1(
λλ
λ
λλ
λ
ωω
ccc
(B. 7)
where with
1
1
~
2
1
1
~
)]1()1(41[
−
−−+= v
ρω
11011
/][ mE
τ
λ
ρ
=
and
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
+
+
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
+
=
2
012
~
011
~
012
~
2
012
~
011
~
011
~
~
1
λλ
λ
λλ
λ
v
2
011
c
and are the SCV for the inter-arrival time of orders to the system of each separate input
flow.
2
012
c
At this point, we have converted the two input flows into one. We can apply the procedures in
Appendix A to calculate the expected waiting time in front of each node and subsequently use
equation (27) to obtain the expected sojourn time of a bin at a node.

Publications in the Report Series Research
∗
in Management
ERIM Research Program: “Business Processes, Logistics and Information Systems”
2007
India: a Case of Fragile Wireless Service and Technology Adoption?
L-F Pau and J. Motiwalla
ERS-2007-011-LIS
http://hdl.handle.net/1765/9043
Some Comments on the Question Whether Co-occurrence Data Should Be Normalized
Ludo Waltman and Nees Jan van Eck
ERS-2007-017-LIS
http://hdl.handle.net/1765/9401
Extended Producer Responsibility in the Aviation Sector
Marisa P. de Brito, Erwin A. van der Laan and Brijan D. Irion
ERS-2007-025-LIS
http://hdl.handle.net/1765/10068
Logistics Information and Knowledge Management Issues in Humanitarian Aid Organizations
Erwin A. van der Laan, Marisa P. de Brito and S. Vermaesen
ERS-2007-026-LIS
http://hdl.handle.net/1765/10071
Bibliometric Mapping of the Computational Intelligence Field
Nees Jan van Eck and Ludo Waltman
ERS-2007-027-LIS
http://hdl.handle.net/1765/10073
Approximating the Randomized Hitting Time Distribution of a Non-stationary Gamma Process
J.B.G. Frenk and R.P. Nicolai
ERS-2007-031-LIS
http://hdl.handle.net/1765/10149
Application of a General Risk Management Model to Portfolio Optimization Problems with Elliptical Distributed Returns for
Risk Neutral and Risk Averse Decision Makers
Bahar Kaynar, S. Ilker Birbil and J.B.G. Frenk
ERS-2007-032-LIS
http://hdl.handle.net/1765/10151
Optimal Zone Boundaries for Two-class-based Compact 3D AS/RS
Yugang Yu and M.B.M. de Koster
ERS-2007-034-LIS
http://hdl.handle.net/1765/10180
Portfolios of Exchange Relationships: An Empirical Investigation of an Online Marketplace for IT Services
Uladzimir Radkevitch, Eric van Heck and Otto Koppius
ERS-2007-035-LIS
http://hdl.handle.net/1765/10072
From Closed-Loop to Sustainable Supply Chains: The WEEE case
J. Quariguasi Frota Neto, G. Walther, J.Bloemhof, J.A.E.E van Nunen and T.Spengler
ERS-2007-036-LIS
http://hdl.handle.net/1765/10176

A Methodology for Assessing Eco-Efficiency in Logistics Networks
J. Quariguasi Frota Neto, G. Walther, J.Bloemhof, J.A.E.E van Nunen and T.Spengler
ERS-2007-037-LIS
http://hdl.handle.net/1765/10177
Strategic and Operational Management of Supplier Involvement in New Product Development: a Contingency Perspective
Ferrie E.A. van Echtelt, Finn Wynstra and Arjan J. van Weele
ERS-2007-040-LIS
http://hdl.handle.net/1765/10456
How Will Online Affiliate Marketing Networks Impact Search Engine Rankings?
David Janssen and Eric van Heck
ERS-2007-042-LIS
http://hdl.handle.net/1765/10458
Modelling and Optimizing Imperfect Maintenance of Coatings on Steel Structures
R.P. Nicolai, J.B.G. Frenk and R. Dekker
ERS-2007-043-LIS
http://hdl.handle.net/1765/10455
Human Knowledge Resources and Interorganizational Systems
Mohammed Ibrahim, Pieter Ribbers and Bert Bettonvil
ERS-2007-046-LIS
http://hdl.handle.net/1765/10457
Revenue Management and Demand Fulfilment: Matching Applications, Models, and Software
Rainer Quante, Herbert Meyr and Moritz Fleischmann
ERS-2007-050-LIS
http://hdl.handle.net/1765/10464
Mass Customization in Wireless Communication Services: Individual Service Bundles and Tariffs
Hong Chen and Louis-Francois Pau
ERS-2007-051-LIS
http://hdl.handle.net/1765/10515
Individual Tariffs for Mobile Services: Analysis of Operator Business and Risk Consequences
Hong Chen and Louis-Francois Pau
ERS-2007-052-LIS
http://hdl.handle.net/1765/10516
Individual Tariffs for Mobile Services: Theoretical Framework and a Computational Case in Mobile Music
Hong Chen and Louis-Francois Pau
ERS-2007-053-LIS
http://hdl.handle.net/1765/10517
Individual Tariffs for Mobile Communication Services
Hong Chen and Louis-Francois Pau
ERS-2007-054-LIS
http://hdl.handle.net/1765/10518
Is Management Interdisciplinary? The Evolution of Management as an Interdisciplinary Field of Research and Education in
the Netherlands
Peter van Baalen and Luchien Karsten
ERS-2007-047-LIS
http://hdl.handle.net/1765/10537

Detecting and Forecasting Economic Regimes in Multi-Agent Automated Exchanges
Wolfgang Ketter, John Collins, Maria Gini, Alok Gupta and Paul Schrater
ERS-2007-065-LIS
http://hdl.handle.net/1765/10594
Emergency Messaging to General Public via Public Wireless Networks
P.Simonsen and L-F Pau
ERS-2007-078-LIS
http://hdl.handle.net/1765/10718
Flexible Decision Control in an Autonomous Trading Agent
John Collins, Wolfgang Ketter and Maria Gini
ERS-2007-079-LIS
http://hdl.handle.net/1765/10719
Discovering the Dynamics of Smart Business Networks
L-F Pau
ERS-2007-081-LIS
http://hdl.handle.net/1765/10732
Performance Approximation and Design of Pick-and-Pass Order Picking Systems
Mengfei Yu and René de Koster
ERS-2007-082-LIS
http://hdl.handle.net/1765/10733
∗
A complete overview of the ERIM Report Series Research in Management:
https://ep.eur.nl/handle/1765/1
ERIM Research Programs:
LIS Business Processes, Logistics and Information Systems
ORG Organizing for Performance
MKT Marketing
F&A Finance and Accounting
STR Strategy and Entrepreneurship