Managing a service system with social interactions: Stability and chaos

Computers & Industrial Engineering 63 (2012) 1178–1188

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Managing a service system with social interactions: Stability and chaos Xuchuan Yuan ⇑, H. Brian Hwarng Department of Decision Sciences, NUS Business School, National University of Singapore, 15 Kent Ridge Drive, Singapore

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 17 May 2011 Received in revised form 22 June 2012 Accepted 25 June 2012 Available online 7 September 2012

This paper investigates the dynamic behavior of a service system in terms of the arrival rate in the steady state under the influence of social interactions. Customers are backward looking and rational when making purchasing decisions. Existing customers’ re-purchasing decisions are based on their experienced utility – a function of the average waiting time and their expected utility. Potential customers are attracted through social interactions with existing customers. It is shown that the arrival rate of the system in the steady state can exhibit stability, periodic cycles, or chaos due to the effect of social interactions and customers’ purchasing behavior. Two examples based on an M/M/1 queueing system illustrate the role of social interactions and the effect of service rates on the stability of the arrival rate in the steady state. The result highlights the dynamical complexity of a simple service system under the impact of customers’ behavioral factors, or social interactions. It suggests a new perspective to managing service operations whereby social interactions may play a critical role in the fluctuations of demand. Ó 2012 Elsevier Ltd. All rights reserved.

Keywords: Service system Queueing system Chaos Social interaction

1. Introduction There are many factors impacting customers’ purchasing behavior in a service system that is repeatedly adopted. The demand for the service is reflected in the fluctuations of the queue of the system and can be measured by the arrival rate. For the managers of the system, fluctuations in arrivals may pose difficulties in capacity and resource management. For the customers of the system, purchasing decisions are not only dependent on the factors inherent in the service, such as the price and the quality, but also on the factors such as the past purchasing experience, the rational expectation, and the influence of other customers through social interactions. Customers’ past purchasing experience and expectation jointly and directly determine if the service is worthy and likely to be re-purchased. On the other hand, social interactions indirectly influence a customer’s purchasing decision. This is especially true for a new customer whose purchasing decision has no prior experience for reference. The above-mentioned behavioral factors can potentially render the arrival process of a service system into a complex dynamical system. It is so even if the service system is run under a deterministic setting, e.g., a constant service rate, price or fixed service quality. The dynamical nature of the arrival process complicates service operations management. The system may turn chaotic. Thus, it is of value to investigate the stability of the arrival process as well as the impact of the underlying behavioral factors

⇑ Corresponding author. Tel.: +65 91379995; fax: +65 68733352. E-mail addresses: (H. Brain Hwarng).

[email protected]

(X.

Yuan),

[email protected]

0360-8352/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2012.06.022

in a service system. Chaos theory is appropriate for the purpose of this study. Pioneered by Becker (1974), social interactions are defined as ‘‘particular forms of externalities, in which the actions of a reference group act on an individual’s preferences’’ (Scheinkman, 2004). The impact of social interactions on customers’ purchasing behavior has been widely studied in marketing (Bearden & Etzel, 1982) and economics (Becker, 1991), while few studies have investigated the role of social interactions in the dynamic behavior of a service system. The purpose of this study is to shed light on the impact of customers’ behavior and purchasing decisions on the stability of the arrival dynamics in the steady state (in the following section, we use steady state and equilibrium interchangeably). Of particular focus is the effect of social interactions among customers. Based on the dynamical system theory, we analyze the impact of these behavioral factors from a chaos perspective. Specifically, we consider the customers who are backward looking (which will be discussed in detail in the following section) when making re-purchasing decisions, and potential customers are attracted due to the social interactions with existing customers. We demonstrate that the dynamics of arrivals in the steady state can be stable, periodic or chaotic due to customer’s purchasing behavior under the influence of social interactions. The structure of the paper is organized as follows. Section 2 briefly reviews relevant literature. Section 3 discusses the model of the service system with heterogeneous backward looking customers in the context of social interactions. The analysis of the model in terms of stability, periodicity and chaos is also provided. Section 4 illustrates the impact of social interactions or the service rate on the dynamic behavior of the service system based on an

X. Yuan, H. Brain Hwarng / Computers & Industrial Engineering 63 (2012) 1178–1188

M/M/1 queueing model. Section 5 concludes with a summary, managerial insight, and future research directions.

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3. The model 3.1. Model specification

2. Literature review The current research is inspired and motivated by two streams of literature. The first stream is about the stability and chaos of a service system (or queueing system). The second stream is about the equilibrium dynamics of a complex system in the supply and demand framework. In the following section, we briefly review several key papers within each stream. Plenty of studies have focused on the stability of dynamical systems. In particular, stability in service systems analyzed from a chaos perspective is an area of great interest. Chase, Serrano, and Ramadge (1993) analyzed the dynamics of the switching time in two discrete control models of a continuous system under full capacity, i.e., a switched arrival system and a switched server system. Their study shows that the switching time in the switched arrival system under a threshold type policy can be chaotic, while the switched server system is generically periodic. Whitt (1993) demonstrated that the dynamic behavior of the queue length at various nodes in a deterministic multiclass network of queues can be chaotic. Feichtinger, Hommes, and Herold (1994) considered a simple discrete time deterministic queueing model with one server and two queues. In their model, the total arrival rate is constant and equal to the service rate; the server adopts a nonlinear decision rule to allocate the service time in each queue based on the difference of the queue length. They discovered that chaotic dynamics appear in the queue length. Haxholdt, Larsen, and van Ackere (2003) demonstrated that in a simple deterministic queueing system, due to the feedback of the customer and the server, the queue length can exhibit the phenomena of sustained oscillation, mode locking, quasi-periodic behavior, or chaos. Stability or chaos of a service system discussed in the above literature is generally analyzed from the server’s perspective, i.e., the supply; few studies have analyzed the impact of various behavioral factors from the customer’s perspective, i.e., the demand. Nakayama and Nakamura (2004) is one of the early works in this area. They developed a logit model to study the adoption rate of a fashion where a customer’s adoption decision is a discrete choice influenced by social interactions, e.g., the bandwagon effect and the snob effect. Their simulation study shows that, due to social interactions, the adoption rate can be periodic or chaotic under certain conditions. Rump and Stidham (1998) studied the stability and chaos of the price and the arrival rate in equilibrium in a multiperiod service system. Their result shows that a customer’s adaptive expectation can contribute to chaos in the price or the arrival rate dynamics in equilibrium. From economics perspectives, the stability and chaos in a supply and demand system is of great interest. Hommes (1994) investigated the price-quantity dynamics of the cobweb model with adaptive expectations and nonlinear supply and demand curves. He showed that chaotic price dynamics can occur generically, even if both the supply and demand curves are monotonic. Brock and Hommes (1997) focused on the dynamical behavior of the price at the equilibrium in a cobweb type demand–supply model with heterogeneous beliefs of suppliers. They concluded that due to the high intensity of choice to switch predictors, an irregular equilibrium price path can converge to a strange attractor and lead to chaos. Brock, Dindo, and Hommes (2006) generalized the model with forward looking suppliers and showed that forward looking behavior dampens the amplitude of price fluctuation, but local instability of the steady state remains. It is demonstrated that the dynamics of a cobweb type demand–supply model are due to the adaptive, rational or forward looking behavior of suppliers. The above literature provides the theoretical basis for our study.

The service system under consideration is one that provides services that can be repeatedly purchased. To focus on the impact of customers’ purchasing behavior on the dynamic characteristics of the service system, we consider the system operating with a constant price and a given quality level. The service is purchased by heterogeneous customers whose primary concern is the waiting time in the system (queue and service). In other words, customers consider the waiting time as their major deciding factor before entering the service system. Suppose the queue length is not observable and customers have to estimate the waiting time before purchasing (For example, it may depend on their past experiences). This is actually observed in certain service systems. For example, in a call center, customers cannot observe the queue and the waiting time is the major factor affecting customers’ decision to adopt the service or not. The congestion-prone systems (Agnew, 1976), such as highways also belong to this category of service systems. Highway users cannot observe the congestion level (or the queue length) of a certain section of the highway, thus they need to estimate the congestion before entering the highway system. (Note: Of course, with modern technology, the congestion information now can be readily available to highway users.) In this paper, we focus on the dynamic behavior of the arrivals in the service system in the steady state, or equilibrium dynamics (Brock et al., 2006) of the service system. We realize that the steady state in real systems may not exist (except for a short time span, since within a short time span, the arrival rate and service rate may not change). Here, we assume the steady state is an approximation or limiting status of a queueing system and the time span of each period is either long enough for the system to converge to the steady state or short enough for a constant arrival and service rate to be observed. Since the queue is not observable, we assume customers adopt the average waiting time in the steady state as their estimation of the time spent in the system. The settings in this paper are similar to those of Rump and Stidham (1998), in which the delay cost is determined by the average waiting time in equilibrium. Based on the above settings, we assume that customers belong to two groups in each period, either existing customers who have purchased the service in the previous period, or potential customers who have not. Customers gain experienced utility after purchasing the service which only depends on the average waiting time in the steady state. Due to heterogeneity, customers have different expected utilities before purchasing. Customers are backward looking, that is the re-purchasing decision of existing customers is based on the comparison between their experienced utility after purchasing and expected utility before purchasing. Potential customers’ purchasing decision is solely influenced through social interactions with existing customers. The social interaction effect functions as such that existing customers who are satisfied with the service system will help attract more potential customers (positive effect thereafter), while those who are not satisfied will discourage potential customers from purchasing the service (negative effect thereafter). As a side note, in the transient stage of a queue, potential customers may be scared away due to a long queue if they care more about the waiting time. However, under the influence of social interactions, more and more customers may be attracted to join the long queue, instead of the short one. Becker (1991) observes this phenomenon where the queue of a seafood restaurant is long while an adjacent seafood restaurant with similar price and quality still has many empty seats available. For ease of reference, the notations of the key parameters and functions used in this paper are listed as follows:

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k: the arrival rate; l: the service rate; w: the waiting time; Wðk; lÞ: the waiting time function; u: the utility; U (w): the utility function; U: the random expected utility; F (u): the distribution of U; p: the proportion of existing customers who will re-purchase the service; pt ¼ pðpt Þ: the social interaction effect in terms of the re-purchasing proportion at period t; Gðk; lÞ ¼ pðFðUðWðk; lÞÞÞÞ: the social interaction effect in terms of the arrival rate and service rate; ktþ1 ¼ hðkt ; lÞ ¼ kGðk; lÞ: the arrival rate at period t + 1 of the simplified one-dimensional dynamical system; Sf ðxt Þ: the Schwarzian derivative of a one-dimensional dynamical system xtþ1 ¼ f ðxt Þ; @hðkt ; lÞ @ 2 hðkt ; lÞ @ 2 hðkt ; lÞ þ2 : the summation of two 2 @l @kt @ l @kt derivatives from the dynamical system;

 g¼

 t: social interaction intensity. Therefore, given fixed service rate l and arrival rate k, the average time that the customer spends in the system (especially the waiting time in the queue) is w ¼ Wðk; lÞ. To facilitate the analysis, we assume: Wðk; lÞ; 0 6 k < l is a non-negative convex function, strictly increasing in k and decreasing in l respectively given another parameter fixed, i.e.,

@Wðk;lÞ @k

> 0;

@Wðk;lÞ @l

< 0; besides, 8l > 0,

we assume Wð0; lÞ ! 0; limk!l Wðk; lÞ ! 1. As discussed previously, in steady state, customers gain their experienced utility after purchasing which only depends on the average waiting time, i.e., u ¼ UðwÞ. Obviously, u is decreasing in w. We consider 0 6 u 6 u, assuming that customers will receive the service eventually no matter how long they have waited and their experienced utility is bounded above even if there is no waiting time (a coordinate transformation can cope with negative utility, while the general result will remain the same). Therefore, we 2

UðwÞ assume U (w) is decreasing and convex in w, i.e., @UðwÞ < 0; @ @w 2 @w > 0, and limw!0 UðwÞ ¼ u; limw!1 UðwÞ ¼ 0. (We may consider the utility function is decreasing and concave by assuming the marginal disutility increases in w. Here we assume the marginal disutility decreases in w based on the non-negativity of u.) Therefore, all existing customers have the same experienced utility in the same period, since all the customers adopt the waiting time in steady state to calculate the experienced utility. As will be seen in the following section, the heterogeneity of customers is categorized by their expected utilities before purchasing. Due to heterogeneity, customers’ expected utilities in each period before purchasing are depicted by a random variable U to capture the distribution from the perspective of the whole population. Since we consider the experienced utility only depends on the waiting time in steady state, the expected utility can also be related to waiting time. We first define the threshold waiting time as the acceptable waiting time beyond which the customer will not enter the system. Here we consider all the customers’ threshold waiting time is þ1 for the sake of the analysis (e.g., the service is a necessity for daily life). (The general result remains the same for other specific threshold waiting time.) U is non-negative on the support ½0; U, where U is the upper bound of customers’ expected utility, which may depend on the service rate l, since in the steady state, customers may be adaptive with the system, i.e., they anchor their expected utility on the service rate. According to Prospect Theory (Kahneman & Tversky, 1979), for a specific existing customer,

the expected utility at the beginning of each period may depend on the experienced utility in the previous periods. However, from the perspective of the whole population, the distribution of all customers’ expected utilities may not depend on the experienced utility in the previous periods, although its mean and variance may be correlated. To simplify the model, let’s assume the distribution of U does not depend on u. Subscript t denotes the period. Those customers whose experienced utility is greater than the expected utility are satisfied with the service system and will re-purchase the service; while the other customers are dissatisfied and will not purchase the service in the next period. Therefore, the re-purchasing proportion of existing customers at the beginning of period t + 1 is ptþ1 ¼ PrðU < ut Þ ¼ Fðut Þ, where FðÞ is the cumulative distribution function of customers’ expected utility. Besides, we assume u 6 U, i.e., the upper bound of the expected utility is large enough. Thus 0 6 Fðut Þ 6 1 is always satisfied. As discussed before, potential customers are attracted by existing customers due to social interactions. We first define the social interaction effect as the following: satisfied customers will attract potential customers to adopt the service; while dissatisfied customers may discourage potential customers from adopting the service. This is demonstrated by the effect of word of mouth that contributes to 20–50% of all purchasing decisions according a survey described in Jacques, Jonathan, and Ole Jorgen (2010). Since the re-purchasing proportion pt is used to approximate the proportion of existing customers who will re-adopt the service, intuitively, the arrival rate should be positively correlated with pt (the positive social interaction effect) while negatively correlated with 1  pt (the negative social interaction effect). However, it is not necessary that the higher pt, the more arrivals, or the higher 1  pt , the lower arrivals. This may be due to the customer’s rational expectation, social interactions or other factors such as limited customer population which may constrain the actual arrivals. We consider the following situation to define the customer’s rational expectation. On the one hand, if pt is significantly larger, potential customers who perceive that the system is highly efficient, i.e., short waiting time, may anticipate a herding behavior, and therefore choose to not enter the system in order to avoid the potential congestion. Existing customers may behave similarly for the same reason by not adopting the service system during the next period. On the other hand, if 1  pt is significantly larger, potential customers who perceive that the system is inefficient, i.e., long waiting time, may conjecture few customers joining the queue in the next period, and therefore choose to enter the system. Existing customers may behave similarly for the similar rationale. To sum up, we make the following assumption: for large pt or 1  pt , due to social interactions and the customer’s rational behavior, the arrival rate in the next period will be smaller than the current period. Therefore, we define the following nonlinear, non-monotonic function to characterize the social interaction effect pt as pt ¼ pðpt Þ, where pt can be considered as the actual ratio of the arrival rate in the following period. If pt P pt , we consider the positive social interaction effect dominates; otherwise, the negative social interaction dominates. We assume pðpt Þ P 0 is concave with a continuous second order derivative denoted as C 2 (C n means the n-th order derivative is continuous) on the support [0, 1] with pð0Þ ¼ pð1Þ ¼ 0, pI ¼ max06p61 pðpÞ ¼ pðpc Þ > 1, and limpt !0 p0 ðpt Þ ¼ 1; limpt !1 p0 ðpt Þ < 1. The assumption indicates that there exist two critical points pc1 and pc2 , such that 0 6 pðpt Þ < 1; 8pt 2 ½0; pc1 Þ [ ðpc2 ; 1, which indicates when pt 6 pc2 , the role of positive social interaction dominates, since we always have pt P pt ; while if pt P pc2 , the role of negative social interaction dominates. Therefore, the evolution of the simple service system in terms of the arrival rate can be formulated as

ktþ1 ¼ kt ptþ1 ¼ kt pðptþ1 Þ ¼ kt pðFðUðWðkt ; lÞÞÞÞ; kt P 0

ð1Þ

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Proposition 4. For a fixed l, there exist three equilibrium points kI i ; i ¼ 1; 2; 3 of the dynamical system ktþ1 ¼ hðkt ; lÞ.

3.2. Stability, periodic and chaos To simplify the model, we first define Gðk; lÞ ¼ pðFðUðWðk; lÞÞÞÞ, and then define hðk; lÞ ¼ kt Gðk; lÞ. Therefore, the model is simplified further as a one-dimensional dynamical system (Devaney, 1989) in terms of k for a fixed l as

ktþ1 ¼ kt Gðkt ; lÞ ¼ hðkt ; lÞ

ð2Þ

To facilitate the analysis, we first derive the following properties of the function Gðk; lÞ and hðk; lÞ in terms of k or l based on the assumptions of Wðkt ; lÞ; Uðwt Þ; Fðut Þ and pðpt Þ.

 The first equilibrium point kI 1 ¼ 0 is stable.    The second and third equilibrium points satisfy G kI i ; l ¼ 1; I c I k2 < k < k3 .  The second equilibrium point kI 2 is unstable.  The stability of the third equilibrium point kI 3 depends on the following situation: t ;lÞ – If 2 < kt @Gðk jkI < 0, the third equilibrium point kI 3 is locally @kt 3 asymptotically stable.

Proposition 1. For a fixed l; Gðk; lÞ P 0 is concave in fkj0 6 k < lg, and there exists a unique maximizer 0 6 kc < l, such that lÞ kc ; @Gðk; @k

c

@Gðk;lÞ @k

t ;lÞ jkI , the third equilibrium point kI – If 2 > kt @Gðk 3 is unstable. @kt 3

>0

c

t ;lÞ – If 2 ¼ kt @Gðk jkI , the stability of the third equilibrium point @kt 3 I k3 is inconclusive.

< 0 for k > k and Gðk ; lÞ ¼ maxk Gðk; lÞ > 1 ¼ p . for k < For a fixed k; Gðk; lÞ P 0 is concave in fljk < lg, and there exists a I

lÞ lc > k, such that @Gðk; > 0 for k < l < lc , @l @Gðk;lÞ < 0 for l > lc , and Gðk; lc Þ ¼ maxl>k Gðk; lÞ > 1 ¼ pI . @l

unique maximizer

Proof. The result is derived from the assumptions in the previous section. h

I Proof. Obviously, kI 1 ¼ 0 is an equilibrium point. The stability of k1 is obvious from the first order derivative of hðkt ; lÞ, since we have the @hðkt ;lÞ jkt ¼0 @kt

¼ Gð0; lÞ ! 0. Obviously, there exist two  I  c I other equilibrium points kI 2 < k < k3 , such that G ki ; l ¼ 1. Based

derivative as

on the property of Gðkt ; lÞ, we have Proposition 2. The critical value kc is increasing in l, and the critical value lc is increasing in k, that is, kc ¼ kðlÞ; lc ¼ lðkÞ are both increasing with respect to the corresponding argument.

1;

@hðkt ;lÞ jk I @kt 3

@hðkt ;lÞ jk I @kt 2

t ;lÞ ¼ 1 þ kt @Gðk jk I > @kt 2

t ;lÞ ¼ 1 þ kt @Gðk jkI < 1. Therefore, the second equilibrium @kt 3

t ;lÞ point is always unstable. If 2 < kt @Gðk jkt ¼kI , the third equilibrium @kt 3

t ;lÞ point is stable, since j @hðk jkI j < 1. h @kt 3

Proof. Based on Proposition 1, we have for fixed l1 ; l2 and l1 < l2 ; Gðkc ðl1 Þ; l1 Þ ¼ Gðkc ðl2 Þ; l2 Þ ¼ pI . That is, the re-purchasing proportion is pc . Since FðuÞ is strictly increasing, there is unique uc , such that Fðuc Þ ¼ pc . Since uðwÞ is also strictly decreasing, there is a unique wc , such that uðwc Þ ¼ uc . Therefore, we have Wðkc ðl1 Þ; l1 Þ ¼ Wðkc ðl2 Þ; l2 Þ ¼ uc . Suppose kc ðl1 Þ > kc ðl2 Þ, based on the assumption of Wðk; lÞ, we have Wðkc ðl1 Þ; l1 Þ > Wðkc ðl2 Þ; l1 Þ > Wðkc ðl2 Þ; l2 Þ, which is a contradiction. Therefore, we have kc ðl1 Þ < kc ðl2 Þ. The same argument can be used to prove the monotonicity of lc ¼ lðkÞ. h In the following section, we focus on the case of a fixed l, taking l as a parameter of the dynamical system. The case of a fixed k can be analyzed similarly. Proposition 3. For a fixed l, there exists a unique maximizer km , satisfying km > kc , such that when kt 6 km , the dynamical system ktþ1 ¼ hðkt ; lÞ is increasing, and when kt > km , it is decreasing and concave. Proof. The first and second order derivatives of the dynamical system are

@hðkt ;lÞ @kt

t ;lÞ ¼ Gðkt ; lÞ þ kt @Gðk and @kt

respectively. Therefore, when kt 6 c

0, while when kt > k ; Gðkt ; lÞ > 0;

@ 2 hðkt ;lÞ @k2t

t ;lÞ k , @hðk ¼ @kt @Gðkt ;lÞ < 0, @kt m

c

t ;lÞ ¼ 2 @Gðk þ kt @ @kt

2

Gðkt ;lÞ @k2t

t ;lÞ Gðkt ; Þ þ kt @Gðk @kt t ;lÞ and limkt !l @Gðk @kt m

l

1. Thus, there exists a maximizer k < l, such that

> <

l > k > kc

m

and Gðkm ; lÞ þ km @Gðk@kt ;lÞ ¼ 0. When kt P km , due to the concavity of t ;lÞ Gðkt ; lÞ, we have 2 @Gðk þ kt @ @kt

Gðkt ; lÞ þ

t ;lÞ kt @Gðk @kt

2

Gðkt ;lÞ @k2t

@hðkt ;lÞ @kt

< 0, indicating that

m

< 0 and the concavity of hðkt ; lÞ. Thus k

¼

is the

unique maximizer of the dynamical system. h Defining the maximum arrival rate of the dynamical system as kM ¼ maxkt hðkt ; lÞ ¼ hðkm ; lÞ ¼ km Gðkm ; lÞ, we also require kM < l to let the steady state exist.

The relationship between the third equilibrium point kI 3 and the maximizer km depends on the steepness of the function hðkt ; lÞ. Since the stability of the first and the second equilibrium points is determined based on the assumptions of the functional forms, the stability or instability of the third equilibrium point thus may complicate the dynamic behavior of the system. 3.2.1. The case with stable kI 3 Suppose the stability condition of the third equilibrium point kI 3 is satisfied. Thus, when the arrival rate becomes larger, I kI 3 P kt P k2 , the evolution of the arrival rate will be attracted to kI . The evolution of the arrival rate when kt < kI 3 3 depends on the location of kt , that is, if kt is not far away from kI 3 , the evolution of the arrival rate will be attracted to kI 3 ; otherwise, it will be attracted to the region ½0; kI 2  and eventually approaching the first equilibrium point 0, since the model is a discrete dynamical system. From the above observation, the domain of the arrival rate can be readily divided into three disjoint intervals:  Region I: fkt j0 6 kt < kI 2 g. In this region, the arrival rate will be I I eventually trapped to kI 1 ¼ 0, since k2 is repelling and k1 ¼ 0 is stable.  Region II: fkt jkI where hðk4 ; lÞ ¼ kI and 2 6 kt < k4 g 2 I m k4 > k3 ; k4 > k . In this region, the arrival rate will be attracted to kI 3.  Region III: fkt jk4 6 kt 6 kM g. In this region, the arrival rate will be attracted to Region I first and then trapped to kI 1 ¼ 0, since the next step of the evolution of the arrival rate will be in Region I and the evolution of the arrival rate will be eventually trapped to kI 1 ¼ 0. Therefore, when the stability condition of the third equilibrium point is satisfied, we note that the dynamic behavior of the system in terms of the arrival rate in the steady state is more predictable, I i.e., the system will eventually converge to kI 1 or k3 depending on the initial point.

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3.2.2. Bifurcation at kI 3

derivative and the value of g, we have the following result (see

t ;lÞ If kt @Gðk jkI ¼ 2, the stability of kI 3 is inclusive. To further @kt

Lorenz, 1989, p. 111):

3

establish whether it is stable or not, we first introduce Schwarzian 3

derivative at a point x of a C -continuous one-dimensional map xtþ1 ¼ f ðxt Þ. Definition 1. The Schwarzian derivative is defined as Sf ðxt Þ ¼  00 2 f 000 ðxt Þ ðxt Þ  32 ff 0 ðx . f 0 ðxt Þ tÞ To facilitate the analysis, we derive the following nth order derivative of the dynamical system as

@ ðnÞ hðkt ; lÞ ðnÞ @kt

¼n

@ ðn1Þ Gðkt ; lÞ ðn1Þ @kt

þ kt

@ ðnÞ Gðkt ; lÞ ðnÞ

@kt

;

nP1

ð3Þ

Based on the assumption, the dynamical system ktþ1 ¼ hðkt ; lÞ in this paper is C 3 -continuous one-dimensional map, thus, the Schwarzian derivative denoted as Shðkt ; lÞ exists as

0 @Gðk ;lÞ 12 @ 2 Gðkt ;lÞ t 3 @2 @kt þ kt @k2t A Shðkt ; lÞ ¼  t ;l Þ 2 Gðkt ; lÞ þ kt @Gðkt ;lÞ Gðkt ; lÞ þ kt @Gðk @kt @kt 3@

2

Gðkt ;lÞ @k2t

þ kt @

3

Gðkt ;lÞ @k3t

ð4Þ

 I  At the third equilibrium point kI 3 , we have G k3 ; l ¼ 1; t ;lÞ jkI ¼ 2, thus the Schwarzian derivative Shðkt ; lÞ evaluated kt @Gðk @kt 3

at kI 3 is 0 ShðkI 3;

lÞ ¼ @3

@ 2 Gðkt ; lÞ @k2t

@ 3 Gðkt ; lÞ 3 @Gðkt ; lÞ @ 2 Gðkt ; lÞ 2  kt  þ kt 3 2 @k @kt @k2t t

!2 1 Aj

kI 3

ð5Þ

Based on the value of the Schwarzian derivativeat kI 3 , we have the following result according to the dynamical system theory (see Lorenz, 1989, p. 137): @Gðkt ;lÞ Theorem 1. At the equilibrium point kI jkI ¼ 2, 3 , suppose kt @kt 3 the following hold: I  If ShðkI 3 ; lÞ < 0, the equilibrium point k3 is stable. I  If Shðk3 ; lÞ > 0, the equilibrium point kI 3 is unstable.

When the equilibrium point of a dynamical system changes from stable to unstable, a bifurcation occurs. The value of the parameters which the bifurcation occurs at is called the bifurcation value. There are several types of bifurcation for a one-dimensional discrete dynamical system. If the first order derivative at the equilibrium point is 1, a flip bifurcation (or period-double bifurcation) as in the above section occurs where two other equilibrium points emerge. Definition 2. The evolution of the dynamical system with the  initial point x is defined as f n ðxÞ ¼ f f n1 ðxÞ with n P 1 and the identity f 0 ðxÞ ¼ x. I We say a branch of period-2 fixed points xI 1 and x2 of the 2 I dynamical system f ðxÞ if xI ¼ f ðx Þ; i ¼ 1; 2, which means i i I I I xI 2 ¼ f ðx1 Þ; x1 ¼ f ðx2 Þ. @hðkt ; lÞ @ 2 hðkt ; lÞ @ 2 hðkt ; lÞ Define another critical value g ¼ þ2 2 @l @kt @ l @kt (see Rump & Stidham, 1998). Based on the above Schwarzian

– 0, there exists a flip bifurcation Theorem 2. If ShðkI 3 ; lÞ – 0; gjkI 3 at the point ðkI 3 ; lÞ. Define the bifurcation value as lc . Depending on the signs of the two values, the following hold:  kI 3 is stable (unstable) for l < lc ðl > lc Þ.  kI 3 is unstable (stable) for l < lc ðl > lc Þ.  A branch of stable (unstable) period-2 fixed points emerge. Therefore, at the bifurcation value, we observe that a period-double bifurcation may occur and one equilibrium point splits into two period-2 equilibrium points. At these new equilibrium points, with the change of the parameter in the system, we may have new bifurcations, and Theorem 2 can be used to determine the stability of the equilibrium point with period 4. The bifurcation may continue and generate the equilibrium point with period 2n ; n ¼ 1; 2;   . Typically, with a continual period-doubling bifurcation, the arrival rate will begin to oscillate in an aperiodic fashion, which is defined as deterministic chaos. The route to chaos is called period-doubling to chaos. 3.2.3. Period-doubling to chaos Different from random or stochastic phenomenon, chaos is generated from deterministic dynamical systems. Chaos can usually be characterized by the following characteristics: sensitive to initial conditions, topologically mixing, and dense of periodic points. For a one-dimensional dynamical system, the existence of chaos can be determined by the Li-Yorke theorem (Li & Yorke, 1975) illustrated in the following theorem: Theorem 3 (Li-Yorke). Let J be an interval and let f : J ! J be continuous. Assume there is a point a 2 J for which the points b ¼ f ðaÞ; c ¼ f 2 ðaÞ; d ¼ f 3 ðaÞ satisfy d 6 a < b < c or d P a > b > c, then 1. for every k ¼ 1; 2;   , there is a periodic point in J having period k; 2. there is an uncountable set S  J (containing no periodic points), which satisfies the following conditions: (a) for every p; q 2 S with p – q; lim supn!1 jf n ðpÞ  f n ðqÞj > 0; lim inf n!1 jf n ðpÞ  f n ðqÞj ¼ 0. (b) for every p2S and periodic point of q 2 J; lim supn!1 jf n ðpÞ  f n ðqÞj > 0. If a one-dimensional dynamical system satisfies the above condition, it is said that the system is chaotic in the Li-Yorke sense. Thus, if a dynamical system possesses a period-3 cycle, the Li-Yorke theorem is satisfied and the dynamical system is in Li-Yorke chaos. To prove the sufficient condition for the existence of chaos in the Li-Yorke sense, we take the following procedure:  The preimage of the maximizer km can be found as the solution of km ¼ hðkt ; lÞ. Due to the shape of the dynamical system, there are two preimages of the maximizer. We choose the smaller one denoted as ksm . Due to the monotonicity of the dynamical system when kt 6 km , we know ksm < km .  The evolution of the maximum value kM is denoted as kMn ¼ hðkM ; lÞ. Thus, the sufficient condition for the existence of period-3 cycle is

kMn 6 ksm < km < kM

ð6Þ

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where 0 6 xt < 1. Therefore, we can find that under the above settings, l has no impact on the behavior of the dynamical system. The critical factor which determines the dynamical behavior is the parameter in the social interaction function form, the intensity of the social interaction. To satisfy the assumptions in Section 3, the following constraints are added to define a proper interval of t : xM ¼ maxt!1 xt < 1; max06x<1 txð1  xÞ ¼ txm ð1  xm Þ > 1; xm < xM . Thus, we obtain the value of t in the range of 92 < t < 27 to represent 4 the social interaction intensity.

Fig. 1. Illustration of the locations of particular values of k. The equilibrium points are k1 , k2 and k3 . km is the maximizer of the dynamic system, and kM is the maximum point. km ¼ hðksm ; lÞ; kM ¼ hðkm ; lÞ; kMn ¼ hðkM ; lÞ; kMn < ksm < km < kM . Therefore, the sufficient condition of Li-Yorke theorem is satisfied and chaos will appear in the dynamical system.

4.1.2. Stability and chaos The three equilibrium points are conveniently derived at xI 1 ¼ 0; pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 I tþ t2 4t, where the first order derivatives at x2 ¼ t 2tt 4t ; xI 3 ¼ 2t pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 2 each equilibrium point are 0; 3  t t2 4t ; 3  tþ t2 4t. Within pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , we get t þ t2  4t and t þ t2  4t are the domain 92 < t < 27 4 increasing, since the derivatives with respect to t are ðt2Þ ðt2Þ pffiffiffiffiffiffiffiffiffiffi  1 > 0; pffiffiffiffiffiffiffiffiffiffi þ 1 > 0 respectively. Thus, we have 1 < 3 2 4t tp t2 4t ffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi t t2 4t < 3 333 ; 3 333 < 3  tþ t2 4t < 1. 2 2 8 8 Therefore, the second equilibrium point is not stable. To determine the stability of xI 3 , we solve the following function pffiffiffiffiffiffiffiffiffiffi t þ t2 4t ¼ 1, and we get t ¼ 16 . To determine the dynamic 3 2 3 , we calculate the Schwarzian behavior of the system when t ¼ 16 3 I derivative denoted as Sf ðxt Þ at xI 3 and get Sf ðx3 Þ < 0; gjxI < 0. 3

Fig. 1 illustrates the above procedure and the locations of these particular values of k.

, a flip bifurcation occurs with two stable periThus, when t ¼ 16 3 od-2 equilibrium points emerging. Next, we identify the conditions for the existence of chaos. Based on the above dynamical system, we can get the following 3 ð274tÞ

4t values: xm ¼ 23 ; xM ¼ 27 ; xMn ¼ 16t

4. Stability and chaos in an M/M/1 queue

m

the maximizer x ¼

2 , 3

273

. To get the preimage of

we need to solve the cubic function

4.1. An illustrated example of the impact of social interactions

tx2 ð1  xÞ ¼ 23. Using a general standard method, we can solve the

4.1.1. Model specification We consider the following specific conditions and functional forms:

cubic function to get three roots. For the above three roots, if the smaller one of the positive roots, denoted as xsm , satisfies the condition of xsm P xMn , the Li-Yorke condition is satisfied. Through some tedious calculation, we find some values of t in the domain satisfy the constraint. Therefore, we have the following result:

 The average waiting time in the steady state is wt ¼ Wðkt ; lÞ ¼ 1 lkt .  The utility function is ut ¼ Uðwt Þ ¼ w1t ¼ l  kt .  We assume the expected utility is uniformly distributed in ½0; U t , thus the re-purchasing proportion is ptþ1 ¼ Fðut Þ ¼ Uut ¼ lUkt . t

t

 We adopt a multiplicative form of pt in terms of pt to characterize the social interaction effect as pt ¼ tpat ð1  pt Þb . Therefore, the social interaction effect is characterized by  a  b ptþ1 ¼ t lUkt 1  lUkt , where t is used to reflect the social t

t

interaction intensity. The larger the value of t is, the stronger the social interaction effect is. To simplify the model, let a ¼ b ¼ 1, thus the social interaction effect can be readily represented by the widely used Logistic equation. Suppose, in the steady state, customers areadaptive with the system, i.e., they are familiar with the efficiency level of the system, so that their expected utility depends on the system itself (we will provide another example where the role of l does impact the dynamic behavior of the model later). Here we consider U t ¼ l.   Thus, the model is simplified to ktþ1 ¼ tkt klt 1  klt . To simplify the dynamical system, we define kt ¼ lxt , where xt is the traffic-intensity which represents the proportion of the time when the system is busy. Therefore, the dynamical system is simplified into a onedimensional model in terms of the traffic-intensity as below

xtþ1 ¼ tx2t ð1  xt Þ

ð7Þ

Theorem 4. Based on the social interaction intensity t, the following results on the stability of xI 3 and the dynamic behavior of the system hold: 

9 2



t ¼ 163: a flip bifurcation occurs with two stable period-2 equilibrium points at xI 3;



16 < 3 Mn

x

< t < 16 : xI 3 is stable; 3

t < 274:

¼

there

16t3 ð274tÞ 273

exists sm

6x

2 3

< <

at 4t , 27

least

tI , such that satisfies tx2 ð1  xÞ ¼ 23.

one sm

where x

The Li-Yorke theorem is satisfied and the system is chaotic in the Li-Yorke sense. For example, when t = 6.18, we get the following values xM ¼ 0:9156, xm ¼ 23, xsm ¼ 0:4382, xMn ¼ 0:4375. Thus the Li-Yorke condition is satisfied. The dynamical system will exhibit chaos in the LiYorke sense. Mn However, due to the stability of xI is in the inter1 , if the point x I val of ½0; x2 , the evolution of the dynamical system will be eventually trapped to 0, although the Li-Yorke condition is satisfied. For example, when t = 6.55, we have the following values xM ¼ 0:9704; xm ¼ 23 ; xsm ¼ 0:4183; xMn ¼ 0:1827; xI 2 ¼ 0:1880. Thus, we have xMn < xI , and the evolution of x will be attracted to xI t 2 1 , since it is stable. Under this situation, a state of transient chaos occurs. For the system to exhibit a state of permanent chaos, a critical value t must be specified, such that

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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t  t2  4t Mn 16t3 ð27  4tÞ sm 2 I x2 ¼ 6x ¼ 6x < : 3 2t 273

bifurcation plot with x 0 =0.5 1

pffiffiffiffiffiffiffiffiffiffi t2 4t 2t

0.9

< 0; the mapping of the larg-

0.8

t We have the following: the second equilibrium point xI 2 ðtÞ ¼

is decreasing in t, since est value x

Mn

ð tÞ ¼

@xI ðtÞ 2 @t

16t3 ð274tÞ 273 (

¼

1 ffi  tpffiffiffiffiffiffiffiffiffi t2 4t

is concave in

since

0.7

t 6 81 @ 2 xMn ðtÞ 96 16 ; ¼ 3 tð27  8tÞ < 0: 27 @ t2 t6 4 27

9 < 2 81 < 16

0.6

x

P 0; @xMn ðtÞ 16 ¼ 3 ð81t2  16t3 Þ @t < 0; 27

ð92 ; 27 Þ, 4

81 I 27 Thus, we have xMn ð81 Þ ¼ 0:7119; xI 2 ð16Þ ¼ 0:2709; x2 ð 4 Þ ¼ 0:1809; 16 I 9 1 Mn 27 Mn 9 2 x2 ð2Þ ¼ 3 ; x ð 4 Þ ¼ 0; x ð2Þ ¼ 3, which means there exists one unique critical value tI , such that

(

Mn xI ðtÞ; 2 ðtÞ > x

t < tI : I Mn x2 ðtÞ < x ðtÞ; t > tI

Actually, the numerical result for the critical value can be solved approximately as 6.5433. Therefore, we have the following result: Theorem 5. Within the subset on the domain of t which satisfies the Li-Yorke condition, denoted as R, there exists a unique value tI , such that

0.5 0.4 0.3 0.2 0.1 0 4.5

5

5.5

6

6.5

7

Fig. 2a. Bifurcation plot with respect to t. The figure is drawn based on xtþ1 ¼ txt ð1  xt Þ with the initial point x0 ¼ 0:5, starting with t = 4.5, and increment by 0.001.

 for t 2 ftjt 2 R; t < tI g, the dynamical system is in permanent chaos in the Li-Yorke sense in terms of the traffic-intensity;  for t 2 ftjt 2 R; t > tI g, the dynamical system is in transient chaos in the Li-Yorke sense in terms of the traffic-intensity. 4.1.3. Simulation result The behavior of the above dynamical system is best illustrated with simulation. Based on the extended simulation, we first plot the bifurcation plot of the traffic-intensity with respect to the social interaction intensity. The bifurcation plot is drawn according to the following procedure: from the smallest t ¼ 92, we run the simulation of the model for 1000 time units with an initial value, we eliminate the first 500 transient data points and plot the following 500 data points which are considered converged; thereafter, we increase the value of t by some small step, say 0.001, from the same initial point, and run the model for another 1000 time units, and then repeat the process of plotting the converged values. Recall that, the location of the starting point is critical for the behavior of the model; if the starting point is in Region I or Region III, the converged value of the system will be 0, the first stable equilibrium point (see Section 3.2.1). To illuminate the interesting outcome, we intentionally select an initial point in Region II, x0 ¼ 0:5. Fig. 2a is the bifurcation plot in terms of the traffic-intensity with respect to t with the initial value x0 ¼ 0:5. We also calculate theLyapunov exponent with respect to t as shown in Fig. 2b. According to the theory of chaos, a positive Lyapunov exponent indicates the existence of chaos (Eckmann & Ruelle, 1985). A negative Lyapunov exponent suggests that the dynamical system is stable. At the bifurcation point, the Lyapunov exponent is 0 for periodic cycles. The Lyapunov exponent can be used to further demonstrate whether the dynamical system is stable, periodic or chaotic quantitatively. From the bifurcation plot in Fig. 2a, we can clearly observe the stable region, bifurcation point, period-3 region and chaos region. When 92 < t < 16 , the third equilibrium point is stable, and the traf3 fic-intensity is eventually converged to this stable point. At t ¼ 16 ,a 3 flip bifurcation occurs, and two stable period-2 equilibrium points occur. With t increasing, another flip bifurcation occurs at each branch of the period-2 equilibrium point and we have four period-4 equilibrium points. Actually, the bifurcation value of t can be estimated by Feigenbaum constant (Feigenbaum, 1978). With t

Fig. 2b. The corresponding Lyapunov exponent with respect to t. The figure is drawn based on xtþ1 ¼ txt ð1  xt Þ, starting with t = 4.5, and increment by 0.001. The negative Lyapunov exponent indicates the system is stable. At the periodical point, the Lyapunov exponent is zero. The positive Lyapunov exponent indicates the system is chaotic.

increasing further, successive flip bifurcations occur and we can get 2n period-2n ; ðn ¼ 1; 2;   Þ equilibrium points. When n is large, the behavior of the dynamical system will be aperiodic, thus chaos appears. The existence of period-3 equilibrium point in the bifurcation plot confirms the existence of chaos in the Li-Yorke sense. Fig. 3a, 3b, and 3c are the first return map in terms of the trafficintensity. From the figure, we can clearly observe the difference between periodic cycles, chaos and transient behaviors of the dynamical system. For highly periodic cycles, the return map is composed of finite isolated points (Fig. 3a). For permanent chaos, the return map is composed of infinite isolated points, which is also defined as a strange attractor (Fig. 3b). For transient chaos, the return map is composed of finite different points, and it is eventually trapped into the first equilibrium point (Fig. 3c).

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1

1 0.9

0.9

0.8 0.7

0.8

x (t+1)

x (t+1)

0.6 0.7

0.5 0.4

0.6

0.3 0.2

0.5

0.1 0.4 0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 3a. Periodic cycle of xtþ1 ¼ txt ð1  xt Þ with t = 6.2. The return map for a periodic cycle consists of finite points.

1 0.9 0.8

x (t+1)

0.7 0.6 0.5 0.4 0.3

0.3

0.4

0.5

0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (t)

x (t)

0.2 0.2

0

0.7

0.8

0.9

1

x (t) Fig. 3b. Chaos (the strange attractor) of xtþ1 ¼ txt ð1  xt Þ with t = 6.4. The return map for a strange attractor consists of infinite points.

From the above example, we can clearly observe the impact of the social interaction effect on the dynamics of the system: with the increase in the level of social interactions, the stability of the steady state may be stable, periodic, or chaotic. It is not difficult to explain this phenomenon: when the interactions among customers are intensified, the system is subject to more and more stimulation or disturbance even if it is very small – a typical characteristic of chaos. The result here may provide an alternative explanation for the observed fluctuations of average demand and the sudden change of mean arrival rates in real systems: the social interactions among customers may be changed. When the effect of social interactions is significant, managers should exert more effort on exploiting customer’s purchasing behavior, rather than merely allocating more resources in raising the service rate. 4.2. An illustrated example of the impact of the service rate In the above example with given conditions, it is noted that the only factor that impacts the dynamic behavior of the system is

Fig. 3c. Transient chaos of xtþ1 ¼ txt ð1  xt Þ with t = 6.6. The return map is eventually converged to 0 (the first equilibrium point which is stable).

social interaction, while the role of the service rate becomes phantom hidden in customers’ adaptiveness (i.e., a customer’s expected utility derived from one’s perception of the service rate). However, as analyzed in Section 3, the impact of the service rate on the dynamic behavior of the system may still be significant. Thus, an example based on the same M/M/1 queueing system is used to explicitly illustrate how the service rate impacts the arrival dynamics. We keep all the functional forms the same as those in the above example, except the re-purchasing proportion function. Instead, we assume customers are not adaptive with the following re-purchasing proportion function: ptþ1 ¼ Fðut Þ ¼ 1  expðut Þ ¼ 1 expðkt  lÞ, i.e, the distribution of the customer’s expected utility depends on the service rate. We also assume the function of social interaction takes the form of pt ¼ tpt ð1  pt Þ by fixing the social interaction intensity t > 4. Thus the dynamical system is simplified to ktþ1 ¼ kt t expðkt  lÞð1  expðkt  lÞÞ. Therefore, there exists at least one equilibrium point kI 1 ¼ 0. The other two equilibrium points I (if they exist) satisfy t expðkI i  lÞð1  expðki  lÞÞ ¼ 1; i ¼ 2; 3, whose existence depends on the value of parameter t. Let us assume l P 1. We first determine the proper range of t. Since kt < l, we have 0 < expðkt  lÞ < 1. Thus we have maxkt t expðkt  lÞð1  expðkt  lÞÞ ¼ 4t. To satisfy the assumption of the social interaction effect and the existence of the other two equilibrium points, we require t > 4. However, for a large enough t, the evolution of kt will eventually converge to kI1 ¼ 0 or kt < 0. The reason is as follows. Suppose for a fixed l P 1, we first assume the starting point k0 is far less than l. Therefore, we will have t expðk0  lÞð1  expðk0  lÞÞ < 1, and k1 < k0 . Based on the assumption of the social interaction effect, we will always have ktþ1 < kt . Thus, the evolution of kt will converge to kI 1 ¼ 0. For a fixed l P 1, assume the starting point k0 is approaching l. We will also have t expðk0  lÞð1  expðk0  lÞÞ < 1, and k1 < k0 . At some stage t P 1, we will have the following situation: kt < kt1 ; t expðkt  lÞð1  expðkt  lÞÞ > 1, t expðkt1  lÞð1  expðkt1  lÞÞ < 1. For a large t, we may have ktþ1 ¼ kt t expðkt  lÞð1 expðkt  lÞÞ P l. If ktþ1 ¼ l, the evolution of kt will eventually converge to kI 1 ¼ 0, while if ktþ1 > l, the evolution of k will be k < 0. The situation for the starting point of a median k0 is similar to that for a large enough t. Therefore, we set a median value of t, say t = 5.5 (which is in the region with period-2 equilibrium points in Section 4.1), to guarantee the existence of the second and third

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return map for μ=1.65

bifurcation plot under =5.5 with respect to μ 1.6

2 1.8

1.5

1.6 1.4

1.4

1.3

λ (t+1)

λ

1.2 1

0.8

1.2 1.1

0.6 0.4

1

0.2

0.9

0 1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

0.8 0.8

2

0.9

1

1.1

Fig. 4a. Bifurcation plot with respect to l increment by 0.001 based on ktþ1 ¼ kt t expðkt  lÞð1  expðkt  lÞÞ with t = 5.5 and k0 ¼ 0:8.

1.2

1.3

1.4

1.5

1.6

λ (t)

μ

Fig. 5a. Periodic cycle of the dynamical system ktþ1 ¼ kt t expðkt  lÞð1 expðkt  lÞÞ with t = 5.5, l = 1.65. The return map for this periodic cycle consists of finite points.

return map for μ=1.8 2 1.8

λ (t+1)

1.6 1.4 1.2 1 0.8

0.8 Fig. 4b. The corresponding Lyapunov exponent of the dynamical system ktþ1 ¼ kt t expðkt  lÞð1  expðkt  lÞÞ with t = 5.5.

equilibrium points and deliberately to reduce the impact from t (the simulation result for the stable region, say t = 5 is similar, but not obvious). Thus, the level of social interactions is not high. The first equilibrium point is stable while the second and the third equilibrium points are unstable. To further demonstrate stability and chaos, the dynamic behavior of the system is investigated qualitatively based on the simulation result. Fig. 4a is the bifurcation plot based on the following condition: t ¼ 5:5; k0 ¼ 0:8; l 2 ½1; 2 increment by 0.001. The similar data processing procedure is used. Fig. 4b is the corresponding Lyapunov exponent with respect to l. From the bifurcation plot, we can observe that, as l increases, the third equilibrium point is stable, periodic with successive flip bifurcations occuring with period-2nequilibrium points, chaotic and eventually converging to 0. The existence of a period-3 cycle in the bifurcation plot further demonstrates the arrival rate is chaotic in the Li-Yorke sense. When l increases beyond 1.9, the arrival rate is eventually

1

1.2

1.4

1.6

1.8

2

λ (t) Fig. 5b. Chaos (the strange attractor) of the dynamical system ktþ1 ¼ kt t expðkt  lÞð1  expðkt  lÞÞ with t = 5.5, l = 1.8. The return map of this strange attractor consists of infinite points.

attracted to 0. The reason is similar to the discussion for a large t in the previous example. Fig. 4b further demonstrates the status of the dynamic behavior in the bifurcation plot, i.e., stable, periodic, and chaotic regions with respect to l. Figs. 5 are the first return map in terms of kt with respect to various l. Similar to the above example, Figs. 5a, 5b and 5c depict the dynamic behavior of the high periodic cycles, chaotic behavior, and transient chaos, respectively. The transient chaos is eventually attracted to the first equilibrium point 0, which is stable. The above result shows that the service rate also impacts the dynamic behavior of the system. For managers, the result indicates if the purpose of increasing service rate is to mitigate the fluctuations of the arrival rate, managers may encounter the unexpected result: the system may be stable for some range of the service rate, while it becomes periodic or even chaotic if the service rate keeps increasing.

X. Yuan, H. Brain Hwarng / Computers & Industrial Engineering 63 (2012) 1178–1188

Fig. 5c. Transient chaos of the dynamical system ktþ1 ¼ kt t expðkt  lÞð1  expðkt  lÞÞ with t = 5.5, l = 1.95. The return map is eventually converged to 0 (the first equilibrium point which is stable).

The practical implication here is, investing more (more resources allocated) in the system to mitigate the arrival fluctuations is not necessarily beneficial. As clearly shown, the interaction effect between customers’ purchasing behavior, i.e., customers’ rational expectation, and social interactions may complicate the system dynamics. A high service rate conveys a perception of a high utility, thus leading to a high re-purchasing proportion. However, due to customers’ rational expectation and social interactions, the actual arrival rate may decrease in the subsequent period (s). This may result in fluctuations in the arrivals even if the service rate is kept high. Interestingly, when social interactions and customers’ purchasing behavior are taken into consideration, this study shows that a manager’s decision, e.g., on the service rate, may contribute to the variability of the arrival rate. This provides an alternative explanation for the occurrence of periodic and fluctuated arrivals of a service system. 5. Conclusion The fluctuations in the customer arrivals to a service system can be contributed to many factors. Given a fixed price and a standard level of quality, the waiting time may become a deciding factor for attracting customers. No doubt about it, a competitive price and consistent quality coupled with efficient and responsive service may attract more customers. It is the goal of any service organization. Internally, there is always room for improvement, e.g., making operations more cost-efficient and quality-conscious. Externally, the voices of customers play a critical role. In today’s network economy, how these voices influence the customers’ purchasing decisions is an increasingly important topic for service organizations. Customers’ purchasing behavior is subject to not only their own experiences and rational expectations, but also the influence of other customers, or social interactions. In this paper, we investigate the dynamic behavior of the arrival process in the queue of a service system in the steady state. Customers are backward looking, rational and subject to social interactions in their decision making (for purchasing). Customers’ purchasing behavior under the influence of social interactions turns the arrival process into a nonlinear dynamical system. In this study, we show that the dynamical arrival process, measured in the mean arrival rate,

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may be stable, periodic, or chaotic in the steady state. The result suggests that the variability of the mean arrival rate can be purely driven by demand under the influence of social interactions, customers’ purchasing behavior, or the interaction of these two forces. Two illustrated examples are used to show the impact of the level of social interactions and of the service rate on the arrival dynamics in the steady state. First, keeping the role of the service rate implicit and other conditions fixed, social interactions can cause the arrival process to be in a stable, periodic, or chaotic state. Second, the service rate may indirectly impact a customer’s purchasing decision and in turn contribute to the stable, periodic or chaotic status of the arrival process. As shown in this study, a simple service system can exhibit complex dynamic behavior. It is specially true when the influence of behavioral factors are present. This obviously makes the operations management of a service system more challenging. The managerial implication for service organizations is as follows. On the one hand, managers need to decide how to allocate resources to the service system and implement suitable policies to ensure the system is running stably. Managers also need to understand the underlying causes of the variability in the volume of orders or sales before making critical decisions, e.g., allocating precious resources to increase capacity unnecessarily. On the other hand, managers need to exploit the role and value of social interactions, for example, leveraging on social interactions to maintain the stability of the mean arrival rate, such as retaining a portion of loyal customers. Although the model devised here is highly stylized, we believe it does capture many essential features of many real-life systems. Several limitations of this study, as well as future research directions, are outlined as follows. First, the use of waiting time as the deciding factor for customers’ purchasing the service is meaningful only when other factors are fixed. Ample research opportunities exist for this to be extended to situations with more general conditions. In this paper, transforming waiting time into utility has a broader applicability because for potential customers who have not experienced the service before, using waiting time directly may not be suitable. Second, although other factors such as quality and price are important to the competitiveness of any service operations, the purpose of this paper is to understand the role of waiting time under the influence of social interactions. Therefore, the results must be interpreted in the context of service with a fixed price and a given quality level. Future research may investigate the effect of price, quality or their interaction on purchasing decisions. Another interesting direction is to understand the above dynamics when there are two or more service providers competing with each other under the influence of social interactions. Besides, the functional form of the social interaction effect can be further explored. Finally, the ultimate managerial question to be answered is how to derive a good policy for managing a service system when social interactions are significant. References Agnew, C. E. (1976). Dynamic modeling and control of congestion-prone systems. Operations Research, 24(3), 400–419. Bearden, W. O., & Etzel, M. J. (1982). Reference group influence on product and brand purchase decisions. Journal of Consumer Research, 9, 183–194. Becker, G. S. (1974). A theory of social interactions. Journal of Political Economy, 82(6), 1063–1193. Becker, G. S. (1991). A note on restaurant pricing and other examples of social influences on price. Journal of Political Economy, 99(5), 1109–1116. Brock, W. A., Dindo, P., & Hommes, C. H. (2006). Adaptive rational equilibrium with forward looking agents. International Journal of Economic Theory, 2, 241–278. Brock, W. A., & Hommes, C. H. (1997). A rational route to randomness. Econometrica, 65(5), 1059–1095. Chase, C., Serrano, J., & Ramadge, P. J. (1993). Periodicity and chaos from switched flow systems: Contrasting examples of discretely controlled continuous systems. IEEE Transactions on Automatic Control, 38(1), 70–83. Devaney, R. L. (1989). An introduction to chaotic dynamical systems (2nd ed.). Menlo Park, CA: Benjamin/Cummings.

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