What is the relationship between the arrival rate of tourists at a transport providing firm's depot and the long run expected wait time experienced by individual tourists in this depot? We use a renewal theoretic framework in this note to answer this question. Our analysis yields two results. First, we show that the long run expected wait time is independent of the arrival rate of the tourists. Second, we explain why this long run expected wait time is equal to the average residual life of a particular renewal process.

Amitrajeet Batabyal

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 1 
  
An independence result concerning the arrival rate of and the provision of 
transport to tourists 
  
 
 
Amitrajeet Batabyal  
Department of Economics, Rochester Institute of Technology 
Abstract 
What is the relationship between the arrival rate of tourists at a transport providing firm's depot and the long run 
expected wait time experienced by individual tourists in this depot? We use a renewal theoretic framework in this note 
to answer this question. Our analysis yields two results. First, we show that the long run expected wait time is 
independent of the arrival rate of the tourists. Second, we explain why this long run expected wait time is equal to the 
average residual life of a particular renewal process.
Batabyal acknowledges financial support from the Gosnell endowment at RIT. The usual disclaimer applies. 
Citation: Amitrajeet Batabyal, (2009) ''An independence result concerning the arrival rate of and the provision of transport to tourists'', 
Economics Bulletin, Vol. 29 no.1 pp. 108-115. 
Submitted: Dec 15 2008.   Published: February 16, 2009. 
 
     1
The reader should note that our subsequent analysis does not depend on the specific means of transport.
1
1. Introduction
There is no denying the fact that tourism is now one of the world’s largest growth industries.
The recent work of Olson (2003) and Batabyal (2007) tells us that tourists from all over the world
generally place a high premium on visiting naturally attractive sites such as fiords, lakes, and
wildlife reserves. It is not standard practice for individuals wishing to tour these kinds of naturally
attractive places to arrange for their own transport. Instead, private firms—and sometimes public
entities—provide the pertinent transportation services. The reader will understand that these kinds
of private firms also frequently provide transport to tourists interested in visiting sites within cities
such as aquariums, museums, and zoos. 
Research conducted by Nordstrom (2005) and by Chaudhary and Batabyal (2008) shows that
the demand for visits to locations that are tourist attractions is probabilistic. This means that a firm
providing transport to tourists is typically operating in a stochastic environment. Hence, a key
problem confronting such a firm concerns the provision of transport. For concreteness, assume that
our transport providing firm uses buses
1 to operate its tours to a particular location. Further, suppose
that these buses carrying tourists to their chosen destination leave our firm’s depot at fixed points
in time. To use the language of Chaudhary and Batabyal (2008), this firm schedules its buses by
time. 
Now, issues pertaining to the provision of transport in general and to the provision of
transport to tourists in particular have been studied by several researchers. Dessouky et al. (2003)
have analyzed an optimization model and have shown that with only marginal increases in operating
costs and service delays, it is possible to cut down the environmental effects of transportation
appreciably. Louca (2006) has empirically studied the long run relationships between, on the one
hand, income derived from the tourism industry in Cyprus and tourist arrivals, and, on the other,
three categories of what he calls supply side expenditures. Focusing on Mauritius, Khadaroo and
Seetanah (2007) point out that the demand for visits to this island depend greatly on the island’s land
and air transport infrastructure. Wolfler Calvo and Colorni (2007) have used a heuristic method to
analyze an emerging transport system in which a fleet of vehicles, without fixed routes and
schedules, carries individuals from a desired pickup point to the desired delivery point. Batabyal
(2008) has studied the scheduling problem faced by a firm providing transport to tourists visiting
a particular location during the slack season. Wang (2008) has used integer programming to
optimize the location and the number of battery exchange stations for electric scooters that can
reduce the environmental impacts of tourism transport. Batabyal and Beladi (2008) have conducted
a stochastic analysis of visitor classification, vehicle breakdowns, and the provision of transport for
tourism. Finally, Chaudhary and Batabyal (2008) have conducted a probabilistic analysis of two
ways of providing transport to tourists.
The papers we have just discussed in the previous paragraph have certainly increased our
understanding of several aspects of the provision of transport in the context of tourism. Even so, to2
See Ross (1996, pp. 98-162) and Tijms (2003, pp. 33-80) for textbook expositions of renewal theory.
3
See Ross (1996, pp. 59-97) and Tijms (2003, pp. 1-32) for textbook discussions of the homogeneous Poisson process.
2
the best of our knowledge, the extant literature has paid virtually no theoretical attention to the
following important question concerning the provision of transport to tourists. What is the
relationship between the arrival rate of tourists at a transport providing firm’s depot and the long run
expected wait time experienced by individual tourists in this depot? Given this lacuna in the
literature, we use a renewal theoretic framework
2 in this note to answer this question. Our analysis
yields two results. First, we demonstrate the counterintuitive result that the long run expected wait
time is independent of the arrival rate of the tourists. Second, we explain why this long run expected
wait time is equal to the average residual life of a particular renewal process. 
The rest of this note is organized as follows. Section 2.1 describes the renewal-reward model
and the renewal-reward theorem in particular. This theorem will form the centerpiece of our
subsequent determination of the long run expected wait time in section 2.3. Section 2.2 uses renewal
theory to describe a simple and parsimonious model in which tourists arrive in accordance with a
homogeneous Poisson process
3 and then wait at our transport providing firm’s bus depot. Section
2.3 first computes the long run expected wait time of an individual tourist in the depot and then
shows that this wait time is independent of the arrival rate of the tourists. Section 2.4 explains why
the expected wait time computed in section 2.3 is equal to the average residual life of a particular
renewal process. Finally, section 3 concludes and discusses ways in which the research in this note
might be extended.
2. The Theoretical Framework
2.1. Preliminaries
The so called renewal-reward model is a very useful tool in the analysis of a whole host of
applied probability models. The thing to note in this context is that many stochastic processes are
regenerative in nature. In other words and as noted by Tijms (2003, p. 33), they regenerate or renew
themselves from time to time so that the behavior of the process after the regeneration or renewal
epoch is a probabilistic reproduction of the behavior of the process starting at time   The time
interval between two regeneration or renewal epochs is called a cycle and the sequence of
“regeneration cycles” makes up a renewal process. What is salient for our purpose is that the limiting
or long run behavior of a regenerative stochastic process on which a particular reward structure is
imposed can be studied in terms of the behavior of the process during a single regeneration cycle.
More formally, the textbook by Ross (1996, pp. 132-137) tells us that a stochastic process 
is a counting process if   represents the total number of counts that have taken place by time 
Clearly, since   etc. are stochastic, the time between any two counts   and 
is also stochastic. This time between any two counts is called the interarrival time. A counting
process for which the interarrival times have a general cumulative probability distribution function
is a renewal process.4
The Poisson process has been used previously by Martin-Cejas (2006), Batabyal (2007), and Chaudhary and Batabyal (2008) to study
problems in the economics of tourism.
3
Consider a renewal process   with interarrival times   which have a
cumulative probability distribution function  In addition, assume that a monetary reward   is
earned when the   renewal is completed. Let   the total reward earned by time   be 
and let   and   The renewal-reward theorem—see Ross (1996, p. 133) or
Tijms (2003, p. 41)—tells us that if   and   are finite, then with probability one,
(1)
In words, equation (1) is telling us that if we think of a cycle being completed every time a
renewal occurs, then the long run expected reward—the left-hand-side (LHS) of equation (1)—is
the expected reward in a cycle or   divided by the expected amount of
time it takes to complete that cycle or   The reader should note that the
renewal-reward theorem describes a long run or steady state result and this theorem holds for
positive rewards such as profits and for negative rewards such as costs. Let us now proceed to our
model of transport provision. 
2.2. A model of transport provision
Consider a transport providing private firm that offers tourists bus tours to a particular
location that is a tourist attraction. These tourists arrive at our firm’s bus depot in accordance with
a homogeneous Poisson process with rate 
4 Buses carrying tourists depart from our firm’s depot
in accordance with a renewal process with interdeparture time   The task before us now is to
compute the long run expected wait time of an individual tourist. To undertake this computation,
we shall use the renewal-reward model described in section 2.1 and, in more detail, in Ross (1996,
pp. 132-149) and in Tijms (2003, pp. 33-80).
2.3. Long run expected wait time
Let   denote the wait time of the   tourist and let   denote the number of tourists
arriving in the depot between the   and the   departure of a bus. Some thought tells us that
the waiting time stochastic process   is a regenerative process that regenerates itself at the
epochs   Now, assume that a reward of   is received for the   tourist. Then,
using the renewal-reward theorem—see section 2.1 above and theorem 3.6.1 in Ross (1996, p. 133)
or theorem 2.2.1 in Tijms (2003, p. 41)—it follows that, with probability one,
(2)4
where   is the expectation operator. 
In words, the long run expected wait time of an individual tourist is equal to a particular ratio
of expectations and that ratio is 
(3)
We can use equation 1.1.10 in Tijms (2003, p. 17) to mathematically describe the expectation in the
numerator of (3). This procedure tells us that 
   In our case, the interdeparture time is   Therefore, the
numerator of (3) reduces to   
Recall that the tourists arrive at our transport providing firm’s depot in accordance with a
homogeneous Poisson process with rate   This fact and the theory of renewal-reward processes
together tell us that the denominator of (3) is given by   Now, putting the results contained
in this and the previous paragraphs together, we get an expression for the long run expected wait
time of an individual tourist in our firm’s depot. That expression is
(4)
Intuitively, we would expect this expected wait time to depend on the arrival rate of the tourists.
However, inspecting equation (4) it is clear that the long run wait time of an individual tourist in our
model is independent of the tourist arrival rate   Mathematically, this result holds because the
arrival rate   appears multiplicatively in the two expectations describing the numerator and the
denominator of (3) and hence cancels out in equation (4). We now proceed to explain why the wait
time in equation (4) is equal to the average residual life of a particular renewal process. 
2.4. Average residual life
In the context of this note, the particular renewal process we are interested in is the renewal
process for which the so called interoccurrence times are given by the interdeparture times of our
transport providing firm’s buses. Further, from Tijms (2003, p. 37) we know that for this renewal
process, the residual life is “the time elapsed from epoch   until the next renewal after epoch  ”
Now, the mean residual life in our model is equal to the long run expected wait time of individual
tourists because the Poisson tourist arrivals occur completely randomly in time. Put differently, this
equality holds because of the well known “Poisson arrivals see time averages” or PASTA property.
This property tells us that the long run fraction of time a stochastic system is in a particular state is5
See Tijms (2003, pp. 53-58) for a textbook exposition of the PASTA property.
5
equal to the long run fraction of tourist arrivals who find the stochastic system in this same state.
5
3. Conclusions
In this note, we used a renewal theoretic framework to analyze the relationship between the
arrival rate of tourists at a transport providing firm’s depot and the long run expected wait time
experienced by individual tourists in this depot. Our analysis yielded two results. First, we
demonstrated the counterintuitive result that the long run expected wait time is independent of the
arrival rate of the tourists. Second, we explained why this long run expected wait time is equal to the
average residual life of a particular renewal process. 
The analysis in this note can be extended in a number of directions. Here are two suggestions
for extending the research described here. First, it would be useful to analyze the extent to which the
results in this note change when the tourists under study arrive at a transport providing firm’s depot
in accordance with a nonhomogeneous Poisson process. Second, it would be instructive to set up and
solve an optimization problem for a transport providing firm in which the long run expected wait time
of an individual tourist from section 2.3 is a part of the objective function being optimized. Studies
of transport provision to tourists in a probabilistic environment that incorporate these features of the
problem into the analysis will provide additional insights into issues in the economics of tourism that
have both theoretical and practical implications.6
References
Batabyal, A.A. (2007) A probabilistic analysis of a scheduling problem in the economics of tourism,
Economics Bulletin, 12, 1-8.
Batabyal, A.A. (2008) Scheduling trips during the slack season: An aspect of the economics of
seasonal tourism. Forthcoming, Tourism Economics.
Batabyal, A.A. and H. Beladi (2008) Visitor classification, vehicle breakdowns, and the provision
of transport for tourism, Letters in Spatial and Resource Sciences, 1, 99-105.
Chaudhary, M.H. and A.A. Batabyal (2008) A stochastic analysis of alternate ways of scheduling
transport for tourists. Unpublished Manuscript, Rochester Institute of Technology.
Dessouky, M., Rahimi, M., and M. Weidner (2003) Jointly optimizing cost, service, and
environmental performance in demand-responsive transit scheduling, Transportation
Research Part D, 8, 433-65.
Khadaroo, A.J. and B. Seetanah (2007) Research note: Does transport infrastructure matter in overall
tourism development? Evidence from a sample of island economies, Tourism Economics, 13,
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Louca, C. (2006) Income and expenditure in the tourism industry: Time series evidence from Cyprus,
Tourism Economics, 12, 603-17.
Martin-Cejas, R.R. (2006) Tourism service quality begins at the airport, Tourism Management, 27,
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Nordstrom, J. (2005) Dynamic and stochastic structures in tourism demand modeling, Empirical
Economics, 30, 379-92.
Olson, M.S. (2003) Practical traveler; seeking roots, finding them, New York Times, October 5.
Ross, S.M. (1996) Stochastic Processes, 2
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Tijms, H.C. (2003) A First Course in Stochastic Models. Wiley, Chichester, UK.
Wang, Y. (2008) Locating battery exchange stations to serve tourism transport: A note,
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Wolfler Calvo, R. and A. Colorni (2007) An effective and fast heuristic for the dial-a-ride problem,
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A First Course in Stochastic Models.

A probabilistic analysis of a scheduling problem in the economics of tourism,

A stochastic analysis of alternate ways of scheduling transport for tourists. Unpublished Manuscript,

An effective and fast heuristic for the dial-a-ride problem, 4OR: A Quarterly

Dynamic and stochastic structures in tourism demand modeling,

Income and expenditure in the tourism industry: Time series evidence from Cyprus,

Jointly optimizing cost, service, and environmental performance in demand-responsive transit scheduling,

Locating battery exchange stations to serve tourism transport: A note,

Practical traveler; seeking roots, finding them,

Research note: Does transport infrastructure matter in overall tourism development? Evidence from a sample of island economies,

Scheduling trips during the slack season: An aspect of the economics of seasonal tourism. Forthcoming, Tourism Economics.

Stochastic Processes, 2 nd edition.

Tourism service quality begins at the airport,

Visitor classification, vehicle breakdowns, and the provision of transport for tourism,

An independence result concerning the arrival rate of and the provision of transport to tourists

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An independence result concerning the arrival rate of and the provision of transport to tourists

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