Taking Advantage of Partial Customer Flexibility
An Inexpensive Means of Improving Performance
Rhonda Righter
Department of Industrial Engineering, University of California, Berkeley, CA, U.S.A.
Keywords: Customer Flexibility, Routing, Scheduling, Service Systems, Call Centers.
Abstract: In many service systems with multiple types of customers, providing server flexibility, e.g., by cross-
training servers, is very expensive. On the other hand, there is often inherent flexibility in some of the
customers that is not used by the system. I argue that taking advantage of such flexibility can create a win-
win-win situation, in which overall performance can be greatly improved, and in which both flexible and
non-flexible customers benefit. Moreover, only a small subset of customers needs to be flexible to obtain
nearly the benefit of full flexibility.
1 INTRODUCTION
In many service, production, and traffic systems
there are multiple types, or classes, of customers
requiring different types of “servers,” i.e., different
services, products, or routes. Often, the underlying
infrastructure is expensive, and hence so are the
opportunity costs incurred when servers of one type
are idle while others are congested. This cost can be
reduced by introducing flexible servers that can
serve multiple types of customers, but the cost of
providing this flexibility may be very high. On the
other hand, in many situations a proportion of the
customers may be flexible, i.e., may be willing to
change their type in order to reduce their time
waiting for service, and the infrastructure to take
advantage of this customer flexibility is often
relatively inexpensive.
Consider a call center (in, say, California), which
provides service in both English and Spanish.
Callers currently have the option of pressing “1” for
English and “2” for Spanish, but there are times
when many Spanish speakers, for example, are on
hold while all the Spanish speaking agents are busy,
and yet there are idle English speaking agents.
Because of the training expense, the cost of errors,
and the high turnover of agents, agents are typically
trained to only handle calls in one language. In such
situations, I argue that the call center should add a
“Press 0” option for bilingual customers willing to
have their question answered in either language in
exchange for reducing their waiting time. Note that
this option has a small incremental infrastructure
cost only, because it is taking advantage of
flexibility that is already present in the customers.
There are many other examples of systems with
partial customer flexibility. An example is the
Mobile Millennium project for reducing traffic
congestion at UC Berkeley, in which participating
drivers collect data on current highway speed
through GPS-enabled cell phones. The data is sent to
a central system that provides information back to
the participating drivers for personal use in choosing
alternate routes (http://www.traffic.berkeley.edu/). A
similar application is to communications and
Internet routing, in which some but not all users
have the ability to query alternate routes and use the
shortest. In a make-to-order manufacturing context,
some customers may not care, for example, about
the color of the product they are ordering. Another
application is to national border crossings with
different queues for different nationalities, and
where some customers may have dual citizenship.
Note that the flexibility I am considering is
customer flexibility, not server flexibility. The latter
has received a lot of attention in the operations
research community, and in particular for call
centers (Aksin et al., 2007); (Graves and Tomlin,
2003); (Hopp et al., 2004); (Hopp and van Oyen,
2004) and (Jordan and Graves, 1995). However,
such flexibility is still generally expensive,
particularly in terms of training costs. Customer
flexibility, on the other hand, is often already
present, but may not be exploited, and generally, it is
301
Righter R..
Taking Advantage of Partial Customer Flexibility - An Inexpensive Means of Improving Performance.
DOI: 10.5220/0004923403010304
In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 301-304
ISBN: 978-989-758-017-8
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
inexpensive to take advantage of customer
flexibility.
In earlier research (Akgun et al., 2011; 2012;
2013) we have shown the benefit of partial
flexibility in homogeneous systems, in which
different groups of servers are stochastically
identical. More work needs to be done, in terms of
investigating the performance of, and developing
protocols for, systems with heterogeneous server
stations and user populations.
2 RESULTS FOR
HOMOGENEOUS SERVERS
If flexible customers are given the option to choose
which queue to join based on queue length, they
would clearly join the shortest queue (JSQ)
assuming server stations are homogeneous, service
times are exponential, and the service discipline is
FCFS. My co-authors and I showed the optimality of
JSQ (join-the-shortest-queue) routing in a very
strong, sample-path, sense (Akgune et al., 2011).
The system is modelled as a queuing system with c
parallel multiple-server stations that have
exponentially distributed service times with the
same service rate μ. All of these servers follow a
nonidling but otherwise arbitrary service discipline
(FCFS, LCFS, etc.). Arrivals to the system form an
arbitrary process that is independent of the state of
the system. Some (dedicated) arrivals are obliged to
use a particular station, while others (flexible) have
the ability to use any of the c stations (or, more
generally, they can use an arbitrary subset of size at
least two of the stations. Dedicated arrivals are
equally likely to require any particular station, so the
arrival process is homogeneous across stations. Let
A be the set of arrival points, and let F
⊆
A denote
the time points where a flexible arrival occurs. Note
that F is an arbitrary subset of A.
We used weak majorization and developed a new
approach for coupling potential service completions
to prove the optimality of JSQ (join-the-shortest-
queue) in the sample path sense. We also showed
that when flexible customers follow JSQ, the total
number of customers in the system is stochastically
decreasing in the proportion of flexible customers,
so there is a strong advantage to having customer
flexibility. Note that minimizing the total number in
the system is equivalent to minimizing the mean
waiting time from Little’s law. We also showed that
the waiting time for dedicated customers is
decreasing in the proportion of flexible customers.
That is, the monolingual customers, on average,
benefit from having bilingual customers.
We also considered several practically important
extensions. For example, suppose customers may
abandon, but they only abandon from the queue (a
reasonable assumption for, e.g., a call center), and
suppose the abandonment rate is greater than the
service rate. We showed, under these abandonment
assumptions, that JSQ no longer minimizes the
number of customers in the system, but it still
maximizes the service completion process. Other
extensions that we considered included finite
buffers, resequencing, random yields, and randomly
varying service rates.
While in Akgun et al., (2011) we showed that
stationary waiting time is stochastically decreasing
in the proportion of flexible customers, p, in Akgun,
Righter, and Wolff (2012) we studied the marginal
impact of customer flexibility, that is, the convexity
of waiting time in p. Convexity means that the
marginal advantage of flexibility is largest at small
proportions. That is, roughly, “a little bit of
flexibility goes a long way.” Unfortunately, it is not
possible to obtain convexity in the strong sense for
which monotonicity holds. We considered a
modified model, which we called the inventory
model, where we obtained a sample-path convexity
result using majorization of the queue lengths. In
this model, there are two servers and they never idle
but instead build up inventory when no customers
are waiting. This may be reasonable in production
environments where demand is high and where it is
expensive to idle machines, e.g., due to high
backorder costs or server shutdown costs. Although
convexity in p is intuitive for our original model, in
which servers idle when they have no customers, it
does not hold in the same strong sense that
monotonicity holds, and it is surprisingly difficult to
prove. We developed a new approach that combines
marginal analysis with coupling to show that the
stationary mean waiting time is convex in p. We
considered a tagged customer in steady-state that has
lowest preemptive priority relative to the other
customers so that the other customers are unaffected
by the tagged customer. We showed that the
derivative of the stationary waiting time with respect
to p (the marginal value of customer flexibility) can
be expressed in terms of the difference in expected
waiting time between going to the long and the short
queue for the tagged customer. We then showed,
using another coupling argument, that this difference
is decreasing in p.
Now consider a slightly different policy, where
flexible customers “virtually” join all of the queues
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for which they are eligible. Once a flexible customer
enters service at one of the stations, its “virtual
copies” are removed from the other stations. Again,
this should not be hard to implement in many service
systems, such as call centers. Such a policy is
equivalent to JSW (join-the-smallest-work) for
flexible customers. We showed in Akgun et al.,
(2013) that such a policy outperforms JSQ.
We were able to improve overall performance, as
well as performance for flexible customers as a
group and dedicated customers as a group, by
moving from JSQ to JSW routing for the flexible
customers. This leads us to ask whether we can do
even better. Of course, the most efficient (overall)
alternative for handling both flexible and dedicated
customers is to maintain a separate queue for
flexible customers, and to follow an optimal
scheduling policy for each server. That is, to decide
whether a given server should serve a dedicated or a
flexible customer next. We showed in Akgun et al.,
(2013) that the optimal policy, under a range of
fairly general conditions, is DCF (serve-dedicated-
customers-first). This policy indeed outperforms
both JSQ and JSW in terms of minimizing overall
mean waiting time, and is especially good for
dedicated customers. However, it is unfair for
flexible customers, and, unlike JSQ and JSW, is not
incentive compatible for them (it is not the policy
that they would choose for themselves).
Figure 1: Comparison of policies. Total number in system
vs. proportion of flexible customers (p).
In Figure 1 we show simulation results for two
M/M/1 queues with overall traffic intensity ρ = .9,
and where
N
denotes the long run average number
of customers in the system. This is directly
proportional to mean waiting time through Little’s
law. When none of the arrivals are flexible (p = 0),
this system becomes two separate M/M/1 queues
with service rate μ (the upper bound line). On the
other hand when all customers are flexible (i.e. p =
1), we see that under JSW and DCF, the mean
number of customers in the system converges to that
for an M/M/2 queue with each server having a
service rate of μ. In other words, when customers are
fully flexible, the system performance is the same as
the performance for the generally more expensive
system in which servers are fully flexible. The lower
bound is a single-server system with twice the
service rate (M/M/1(2μ)), which would represent an
ideal (and generally unattainable) pooled scenario
where servers can collaboratively serve each
customer with no loss in efficiency.
We see from Figure 1, as expected, that DCF
outperforms JSW, which in turn outperforms JSQ.
Note, however, that the DCF performance is not
much better than that of JSW, and, as mentioned
before, it is unfair to flexible customers. Therefore,
the best overall policy is JSW. The figure also
clearly shows the convexity of performance. In
particular, we have an “80-20 rule” where at about p
= 20% we have about 80% of the benefit relative to
the total benefit that could be obtained by going
from p = 0 to p = 1.
3 FUTURE WORK
Results for homogeneous stations clearly indicate
the benefit of exploiting customer flexibility. Much
work remains to be done to study systems with
heterogeneous servers and multiple classes of
customers.
REFERENCES
Akgun, O., Righter, R., and Wolff, R., 2011. Multiple
Server System with Flexible Arrivals. Advances in
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Akgun, O., Righter, R., and Wolff, R., 2012.
Understanding the Marginal Impact of Customer
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Akgun, O., Righter, R., and Wolff, R., 2013. Partial
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