On the Pooling of Queues: How Server Behavior Affects Performance

On the Pooling of Queues: How Server Behavior Affects Performance Hung Do School of Business Administration, University of Vermont, [email protected]

Masha Shunko Krannert School of Management, Purdue University, [email protected]

Marilyn T. Lucas School of Business Administration, University of Vermont, [email protected]

David Novak School of Business Administration, University of Vermont, [email protected]

It is widely accepted that multi-server single-queue (SQ) systems outperform multi-server parallel-queue (P Q) systems due to the pooling effect. However, humans are not machines, and the assumption that servers perform identically regardless of the queueing system, is not necessarily true in practice. We investigate the impact of two human server behaviors - social loafing and workload dependent speedup - and one physical factor - walking time - on the performance of the multi-server SQ and P Q systems. We develop theoretical models for these queueing regimes and comparatively analyze them with respect to three performance indicators: expected wait time in queue, pre-service time, and total system time. In addition, we propose functional forms for social loafing and speedup effects and evaluate the performance of the models numerically. On average, social loafing in SQ worsens the performance of the system. In P Q, the benefit of server speedup on system performance is positive and, moreover, is greater than the direct impact of a comparable average increase in the server’s speed. For all three performance measures, we derive threshold values beyond which P Q systems outperform SQ systems. For a set utilization, when the speedup effect is low to medium relative to social loafing, P Q systems dominate SQ systems for small and large numbers of servers. Published research considering the interactive effects of human behaviors on queueing system performance has been scarce; we believe that our theoretical formulations and numerical insights will provide managers with guidance in the design of their queueing operations and serve as building blocks for researchers in their future work on behavioral queueing.

Key words : Service System Design; Single Queue System; Multiple Queue System; Workload-dependent Service Rate; Social Loafing

1. Introduction The management of queues has long intrigued researchers in service operations. Queueing systems are characterized by an arrival process with a rate at which customers or units come into the system, and a service process with a rate at which a particular service is delivered by servers or employees in the system. Queues constitute the buffering mechanism by which differences between the arrival rate and service rate are managed (Rafaeli et al. 2002). The specific structure of the queueing 1

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system is of particular importance in service operations involving human servers. In situations where multiple employees simultaneously provide service to multiple customers, the design of the service delivery system can differ in two fundamental ways: (1) all arriving customers can be combined (or pooled) into one single queue to receive service, or (2) each arriving customer can choose the specific queue in which to wait and consequently the employee from which they receive the service (Rafaeli et al. 2002). These two alternative queueing design structures are referred to as multi-server, single queue (SQ) and multi-server, parallel-queue (P Q), respectively. A classic example of such a design structure decision is the choice faced by a grocery store whether or not to consolidate its multiple checkout lines into a single-line “next available server” checkout model. In 2009, the Hannaford Bros. Company, a New England chain of grocery supermarkets, engaged in an experiment that entailed changing the layout of its checkout process at some of its store locations from a traditional P Q to a SQ configuration (Fantasia 2009). This initiative, spearheaded by Hannaford’s headquarters, was described in the company’s annual report to its investors as a means to enhance operational efficiency and customer experience (Delhaize Group 2012). Queue design has long-term implications on the performance of the store and reduction in wait time has been positively linked empirically to higher financial performance in firms through increased market share and reduced costs (Allon et al. 2011). To the best of our knowledge, there is limited analytical research to date to guide firms in the design of their queueing systems taking into account behavioral and physical considerations. Our research should help managers in making such a decision with long-term implications. Traditional queueing theory allows us to analytically compare the performance of an SQ system with s servers to that of a P Q system with s identical servers, each with its own dedicated queue. It has been shown that enhancing pooling effects by combining queues reduces both the mean and variance in waiting times. Mandelbaum and Reiman (1998) point out that pooling can take several forms (i.e., “pooling queues (the demand), pooling tasks (the process) and/or pooling servers (the resources)”, and, in some instances, lead to mixed performance results. For example, the heterogeneity of demand may negatively affect the pooling benefits in a 2-server system (e.g., van Dijk and van der Sluis (2008), Joustra et al. (2010)). In this paper, the focus is on the pooling of queues. It is widely accepted that due to this pooling effect multi-server, SQ systems outperform multi-server, P Q systems (Smith and Whitt 1981). It is interesting to note, however, that while many banks, airline ticketing and baggage check, and post offices have embraced the SQ structure (Fantasia 2009), its adoption by other service organizations has been more tentative; and the multiserver (P Q) design has remained the norm for many service organizations, especially in the retail and fast-food industries. Thus, despite the theoretical superiority of the SQ structure, it appears

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that combining, or pooling queues might not always be best system design option in practice (Rothkopf and Rech 1987). It has been observed that human-based service queueing systems are not the same as computerized or automated queueing systems, and that models which rely on traditional, automated queueing assumptions, fall short in explaining what is being observed in practice because they lack the “human element.” When traditional queueing models are applied to physical service queues, individual workers are assumed to be identical, and to deliver services at the same rate, just like machines. An implicit assumption in these models is that the behavior of the individual workers, and thus worker productivity, is completely independent of the system design parameters. However, this assumption clearly contradicts a fundamental human resource management and industrial psychology insight that shows employees respond to their environment in non-random ways (Bendoly et al. 2010, Boudreau et al. 2003). Given this, it is not surprising that the use of traditional queueing models have failed to inform observations from real-world service operations. We contend that there is a real need to more effectively incorporate actual human behaviors into the mathematical models used in service operations. As Bendoly et al. (2010) summarize: “We believe that [...] incorporating human behavioral factors into OM models and empirical research will provide gains not only to the practical nature of existing theoretical models but also to the field’s general understanding of what it means to have effective operations.” Despite the fact that Smith and Whitt (1981) provided theoretical proof that SQ systems outperform P Q systems, many managers have been reluctant to combine or pool queues in practice for a variety of reasons. In early papers, Rothkopf and Rech (1987) point to behavioral factors that may undermine the pooling benefits of the single queue while Schultz et al. (1998) show that workers’ processing times are not independent of system design parameters. More recently, Bendoly et al. (2010) identify social loafing and saliency of feedback as having an impact on worker productivity. Shunko et al. (2014) show that these behavioral factors slow the worker’s service speed in SQ systems, making these systems less attractive than theoretical predictions would indicate. Similarly, in P Q systems where congestion is high and highly visible, using experimental and empirical work, research shows that workers tend to “speed up” (Schultz et al. 1998, Shunko et al. 2014). In a recent study of queue configuration in larger queueing systems, (Yanagisawa et al. 2010) note that some walking distances are too large to ignore, and clearly show that walking time, or the time spent walking from the head of a queue to the next available server, does impact the overall performance of the system. H¨am¨al¨ainen et al. (2013) call for incorporating such behavioral phenomena into OR models to improve theory and to obtain new insights. In this paper, we model, formulate, and analyze behavioral factors into queueing models to directly investigate the impact of queue design parameters on system performance. We draw upon

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existing theories from social psychology (Bendoly et al. 2010, Boudreau et al. 2003) and build on insights from recent empirical and experimental studies (Delasay et al. 2015, Shunko et al. 2014) and theoretical models (Yanagisawa et al. 2010) to relax the traditional queueing assumption that employee behavior is independent of system design factors (Bendoly et al. 2010) in an effort to accurately incorporate “real-life” human behaviors into our decision-making models. We conduct a comparative analysis of the performance of SQ and P Q systems, recognizing that alternative queue structures are likely to alter the physical flow of customers throughout the systems, as well as server behaviors within the systems, thus affecting the overall performance of each system. Specifically, we model and then analyze the performance of two queueing regimes (the SQ and P Q systems) when the physical factor of walking time is accounted for in the SQ system, and the behavioral factors of social loafing and workload-dependent speedup are present in the SQ and P Q systems, respectively. We compare the performance of the two systems using three performance measures: 1) expected wait time in queue, 2) expected pre-service time, and 3) expected total time in system - under the assumption of homogeneous servers and demand as well as non-strategic customers. Future work in this area could relax some of these assumptions. It is important to note that conducting a comparative analysis of these two systems is not trivial. Because of the complexities brought about by the workload-dependent speedup in the P Q system, we approach this comparison in the following manner: we first compare the P Q system to an equivalent P Q system without the workload-dependent speedup, and then compare this equivalent P Q system without the workload-dependent speedup to the SQ system. Using this approach, for each performance measure, we derive a threshold performance value beyond which the P Q system outperforms the SQ system. We demonstrate that, in the P Q system, where human servers increase their service rates given higher perceived workload (as measured by the number of customers in the system), the effect of workload-dependent speedup on system performance is above and beyond the effect of a comparable increase in service rate. Namely, the system does not only benefit from the direct impact of an increased service rate, but also benefits from the indirect impact that the workloaddependent speedup has through the smoothing of the queueing process. More precisely, when the arrival rate is held constant, the queue size and number of customers in the system resulting from the speedup effect are stochastically smaller than those resulting from a comparable increase in service rate. To evaluate these effects numerically, we propose to capture the social loafing effect using a functional form that is decreasing convex in the number of servers and to capture the effect of workload using a functional form that is increasing concave in the workload. Our numerical results demonstrate an interesting non-monotone rule for selecting between the SQ or P Q systems. When the speedup effect is strong relative to the pooling and social loafing effects, P Q systems dominate SQ systems. For the medium speedup effect, P Q systems dominate SQ

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systems for either small or large numbers of servers, while SQ systems dominate for situations in between. This observation, along with other decision rules and insights developed throughout the manuscript, provide decision makers, such as e.g. retail stores’ managers, with invaluable guidelines on designing and staffing service systems that result in improved customer experience and better firm performance. There are a number of novel scientific contributions associated with this research. First, to the best of our knowledge, this paper is the first study that theoretically analyzes the impact of behavioral and physical factors on system performance. We model, formulate, and analyze the impacts of these factors on the three performance measures. Second, there are very few sources that consider impact of physical factors such as walking time on system performance. We include the walking time into the queueing model of the SQ system and analyze its impact on the performance of the system. We develop generalized theoretical models that are applicable to both physical and non-physical (computerized or automated) queueing systems, demonstrate the magnitude of these effects using numerical examples, and discuss the practical implications of our findings by presenting generalized decision rules that can guide organizations or managers in selecting the most appropriate service-based queueing system design strategy.

2. Literature review A review of the literature shows that the analysis of traffic delays at toll booths by Edie (1954) was one of the earliest studies to provide evidence of the speedup effects by employees in response to system congestion. Several studies have since demonstrated that the traditional queueing assumption that employee behavior is independent from system design might not always hold true (see Delasay et al. (2015) for a review). Of particular importance is the work by Schultz et al. (1998) who investigate worker motivation in JIT production systems. The authors conclude that processing time distributions in production systems are not independent of factors such as buffer size, processing speed (or working speed) of co-workers, or inventory in the system. They also argue that the magnitude of these factors is significant, and their direction is predictable. Additional evidence suggests that processing times depend on both design and congestion level of the queueing system (Doerr et al. 1996, Kc and Terwiesch 2009, Tan and Netessine 2014), as well as on behavioral elements that reflect workers attitudes and co-workers relationships with each other (Siemsen et al. 2007, Mas and Moretti 2009). Finally, in a review article of recent empirical studies that investigate the influence of load on service times, Delasay et al. (2015) discuss a number of theoretical mechanisms to explain how system characteristics might affect employee behavior and cause service times to change.

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We next introduce the theoretical foundations of the two behavioral factors of interest in this study - social loafing and workload dependent speedup - and discuss the theoretical impacts of each factor on employee processing times. We then articulate how these factors alter the modeling assumptions in both a multi-server SQ and a multi-server P Q environment. 2.1. Social Loafing Effect: Interdependence theory, which is grounded in industrial and organizational (I/O) psychology, has a profound effect on employee motivation. In general, the theory maintains that the way in which goals are structured determines how individuals interact in a particular setting, and this, in turn, shapes outcomes (Johnson and Johnson 2005). The nature of the interdependence between two individuals depends directly on how, and to what extent, each person can affect what happens to the other. In the case where collaborative or joint efforts across employees are necessary to achieve a desired outcome, and where joint rewards exist, the reward to each employee is directly dependent upon the performance of her co-workers. The realization that she might not receive the full benefits of her own “hard work” induces the employee to exert less effort than she might exert if she were being rewarded for performing an individual task (Bendoly et al. 2010). This phenomenon, referred to as social loafing (Latane et al. 1979), has been widely observed in practice (Simms and Nichols 2014). Social loafing is defined by (Karau and Williams 1993) as “the reduction in motivation and effort when individuals work collectively compared with when they work individually,” and has extremely important implications in the context of service queue design choice. Bendoly et al. (2010) argue that changes in the queue design structure will affect the dynamics of interdependence among the collection of servers as a group. Therefore, a shift from P Q to SQ is likely to increase the perception of interdependence among the servers, increase the effect of social loafing, and thus negatively affect the speed at which servers work in the SQ system. Shunko et al. (2014) confirm this hypothesis using data from a behavioral lab experiment. In a meta-analysis of 78 studies, Karau and Williams (1993) find that social loafing is positively correlated to group size (Latane et al. 1979). In the context of SQ service systems, this implies that the speed at which servers work is a decreasing function of the number of servers due to the effect of social loafing. Assumption 1. Social Loafing Effect: In SQ, the service rate depends on the number of servers. 2.2. Workload Dependent Speedup Effect: At the core of control theory models of motivation is the belief that “individuals use feedback to look for differences between their goals and their performance in order to regulate their behavior” (Bendoly et al. 2010). In other words, workers tend to adjust their work rates in response to certain

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perceptions, situations, and incentives. This phenomenon is referred to as state-dependent behavior (Powell and Schultz 2004), and “speedup,” in this context, involves the server working faster or harder, thus effectively increasing her service rate. For example, in Tan and Netessine (2014), facing a high load, or being busy, resulted in restaurant wait staff accelerating the speed at which they provided service to customers. Similarly, a study of patient transporters in a hospital (Kc and Terwiesch 2009) showed that transport staff alter their speed in response to load. According to Schultz et al. (1998, 2003), the speedup effect becomes more evident when feedback about employee performance becomes salient. In a P Q environment, where each server is responsible for her own queue, the size of the individual queue in front of each service delivery station provides a salient feedback mechanism to the employee. In real-world operational settings, supervisors often use queue size as a proxy for the server performance. Thus, if the line for her service station is long, the employee will speed up her processing in order to avoid managerial perceptions of not working hard, or not working efficiently. Moreover, the size of the queue is also clearly visible to other servers, customers, and supervisors in the system, which may put additional pressure on the employees to work faster (Schultz et al. 2003, Bandiera et al. 2013). Correspondingly, servers tend to “speed up” when the number of customers waiting in line is large and the line is visible by others in the system. Assumption 2. Workload Dependent Speedup Effect: In P Q, the service rate depends on the perceived workload determined by the number of customers in the system. 2.3. Walking Time: According to the physical structure of the SQ system, when it is her turn, a customer must walk from the common single queue to the next available server to initiate the service encounter. This initial phase of the service encounter has been described by (Bitran et al. 2008) as the “access” phase. In this paper, we refer to this time spent walking from the head of the single queue to the next available server as the average walking time, and this average walking time is increasing in both distance and number of servers. For example, in a large pooled SQ system such as a ticketing counter or immigration inspection at a large international airport, customers may wait in a single, centralized queue that may be serviced by 20+ service counters. Obviously, some service counters are physically further from the head of the queue than others. The average walking time contributes to the pre-service time component of the expected total time spent by customers in the system. Given that the pre-service time consists of the waiting time in queue (i.e., waiting to be served) and the time spent walking to a server, in such large queueing systems where queue configuration affects walking distances, the effect of walking time cannot be ignored (Yanagisawa

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et al. 2010). Thus, we argue that in large pooled SQ systems, the average walking time could be a non-trivial component of the pre-service time that should be accounted for in performance comparisons between SQ and P Q systems. It is worth noting that, for non-physical queueing systems such as computer processing and call centers, customers send requests to the system and the requests are queued, not the customers, so there is no walking time. Our model, however, is general enough to accommodate both cases.

3. Models and their Performance 3.1. Modeling setup and basic assumptions We consider a service environment, in which customers arrive according to a Poisson process with rate λ. The service time is exponentially distributed with service rate that depends on the structure and parameters of the queueing system as described below. The single queue (SQ) model is represented by an M/M/N queue with a possible social loafing effect on service rate, and the parallel-queue P Q model is represented by N M/M/1 queues with a possible workload dependent speedup effect on service rate. Let L(t) and Q(t) be the number of customers in the system and in the queue at time t (in the P Q model with symmetric servers, the analysis of one line will be representative of the system). Based on Assumptions 1 and 2, the service rate may depend on the system structure (SQ or P Q), the system state (namely perceived workload determined by the number of customers in the system L(t)), and the number of servers N . We use N as a proxy for the system size. When necessary, we subset parameter t. Let si (L, N ) be the service rate where i is the system indicator. Our analysis and derivations can be readily extended to facilitate general functional forms of service rate si (L, N ) where the service rate of a server could depend on both the number of customers in the server subsystem and the number of servers in the whole system. For ease of exposition, we assume the following properties of functional forms: • System SQ0 : Benchmark case with a Single Queue. The service rate is independent of the

queue structure, the workload, and the total number of servers: sSQ0 (L, N ) = µ0 ; • System P Q0 : Benchmark case with Parallel Queues. The service rate is also independent of

the queue structure, the workload, and the total number of servers: sP Q0 (L, N ) = µ0 ; • System SQ: Single Queue (SQ) system where the service rate depends on the number of servers

due to the Social Loafing Effect (Assumption 1). Since the social loafing effect is stronger when the number of servers is larger, sSQ (L, N ) = sSQ (N ) is weakly decreasing in the number of servers N. When there is only one server, there is no social loafing effect and sSQ (1) = µ0 ; • System P Q: System of N parallel M/M/1 queues where the service rate of a human server

depends on the number of customers in her line (Assumption 2), but is independent of the number of

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servers (i.e., there is no social loafing). In this case, each server has her own queue and sP Q (L, N ) = sP Q (L), which is weakly increasing in L(t). In the case of one customer in service, there is no speedup effect and sP Q (1) = µ0 . In addition, we note that in a physical SQ system, customers may spend some amount of time walking from the head of the queue to an available server, while in the P Q system, this time is likely to be negligible. The expected walking time depends on the physical design of a queueing system. To illustrate the concept, we assume a simple design as follows. Let τ be the time it takes a customer to walk from one server to the next, and assume that the SQ head is located in the middle of the range of servers. Let n =

N 2

if N is even and n =

N −1 2

if N is odd. The expected

walking time can then be calculated as follows. Lemma 1. The expected walking time from the common queue to an available server is:  n(n+1) if N is odd, τ 2n+1 t(N ) = n τ2 if N is even. The walking time t(N ) is increasing proportionally to the number of servers and may materially impact performance when the number of servers is large. While the functional form of t(N ) could be different for different physical queueing structures, we assume the functional form in Lemma 1 for illustrative purposes. We compare the performance of the P Q and SQ systems stochastically in terms of the number of customers in the system and the number of customers in queue(s). In addition, we examine the following three time-based expected performance measures (summarized in Figure 1) which, by Little’s Law, are equivalent to expected performance measures based on the number of customers: 1. Expected waiting time in queue; 2. Expected pre-service time, which consists of expected waiting time and walking time from the head of the queue to a server; and 3. Expected total time spent in the queueing system, which consist of expected wait time, expected walking time, and expected service time. Time'spent'by' customer'in'a' queueing'system:' Performance( measures:(

Wai$ng'$me'

Walking'$me'

Expected(Wai,ng(Time( Expected(Pre3Service(Time( Expected(Total(Time(

Figure 1

Performance measures used in our analysis.

Service'$me'

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For the analytical comparison, we assume that customers join lines in the P Q model independently. This assumption is representative of non-physical queueing systems such as service tickets in a technical support group, electronic advising requests in a student office, etc. In these settings, the customers cannot select which line to join and cannot jockey between the lines. In other physical queueing systems, however, customers may behave strategically by comparing the length of all queues and joining the shortest queue (JSQ), by comparing the lengths of a subset of queues before making a joining decision (Supermarket Model, see e.g. Graham (2000)), and/or jockeying between the lines or even balking. Alternatively, customers may act strategically by inferring quality from the length of the line and join longer queues (“herd”), see e.g. Kremer and Debo (2015). Consideration of strategic customers’ behavior is outside of the scope of our study as our research focus is on the servers’ behavior. Consequently, the analytical thresholds for the P Q queueing structure derived further are approximations for real-world P Q systems observed in practice. 3.2. Benchmark for Single Queue: System SQ0 Let traffic intensity be ρ(µ) =

λ µ

and server utilization u(µ) =

λ Nµ

=

ρ(µ) , N

where µ is any given

service rate. For simplicity of notation, we write ρo = ρ(µ0 ) and u0 = u(µ0 ). Let QSQ0 (µ), LSQ0 (µ) be the numbers of customers in queue and in the system when the service rate of each server is µ, where µ is independent of the number of customers and number of servers. Let W SQ0 (µ) and T SQ0 (µ) be the expected wait time in queue and expected total time in the system for a customer, respectively. Let π SQ0 (L, µ) be the stationary distribution of LSQ0 (µ). The standard analysis of this M/M/N system gives: π SQ0 (0, µ) =

N −1 X j=0

π

SQ0

(k, µ) =

ρj (µ) ρN (µ) + j! N !(1 − u)

!−1 ,

( k π SQ0 (0, µ) ρ k!(µ) π

0≤k≤N

N (0, µ) ρ N(µ) uk−N (µ) !

SQ0

The expected number of customers in queue is E[QP0 Q (µ)] =

k > N.

u(µ) C(N, ρ(µ)), (1−u(µ)

where C(N, ρ(µ))

is the Erlang C formula. We can show that: E[QSQ0 (µ)] =

(ρ(µ))N +1 u(µ) SQ0 π (N, µ) = π SQ0 (0, µ), and (1 − u(µ))2 (N − 1)!(N − ρ(µ))2

(1)

the expected number of customers in the system is then: E[LSQ0 (µ)] = E[QSQ0 ] + ρ(µ).

(2)

Correspondingly, using Little’s Law, we derive the expected wait time in queue: E[W SQ0 (µ)] =

N ρN +1 (µ) π SQ0 (0, µ). λ(N − ρ(µ))2 N !

(3)

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We include the expected walking time, t(N ), in the expression of the expected total time in system: E[T SQ0 (µ)] =

1 N ρN +1 (µ) + π SQ0 (0, µ) + t(N ). µ λ(N − ρ(µ))2 N !

(4)

In non-physical queueing systems, there is no walking time, and t(N ) = 0. 3.3. Benchmark for Parallel Queue: System P Q0 For the P Q system, we assume that the Poisson arrival process is split between N servers, where each server receives customers at a rate

λ N

and processes them at a rate µ. For each individual queue

in the system of parallel queues, let QP Q0 (µ) and LP Q0 (µ) be the number of customers waiting in queue and the total number of customers in the system respectively, when the service rate of each server is µ, where µ is independent of the workload and number of servers. Let W P Q0 (µ) and T P Qo (µ) be the expected wait time in queue and the expected total time in the system for a customer, respectively. On average, a customer will face an expected wait time similar to the one experienced in a M/M/1 system and we use standard M/M/1 notation to describe the relationships. Let π P Q0 (L, µ) be the stationary distribution of LP Q0 (µ) with service rate µ. The probability of having no customers in the system is π P Q0 (0, µ) = 1 − u(µ). Assuming that the walking time in P Q is negligible, we set t(N ) = 0. All expected performance metrics are then as follows: u(µ) , 1 − u(µ) u2 (µ) E[QP Q0 (µ)] = , 1 − u(µ) E[QP Q0 (µ)] 1 ρ2 (µ) = , E[W P Q0 (µ)] = λ/N λ N − ρ(µ) E[LP Q0 (µ)] 1 E[T P Q0 (µ)] = = . λ/N µ(1 − u(µ)) E[LP Q0 (µ)] =

(5) (6) (7) (8)

3.4. Single Queue with Social Loafing effect: System SQ We assume that in the SQ system, the social loafing effect decreases the expected service rate of each server. The decrease is captured by the social loafing factor β(N ) =

sSQ (N ) , µ0

where 0 ≤ β(N ) ≤ 1

and β(N ) is weakly decreasing in N . Let π SQ (L) be the stationary distribution of LSQ . Following an analysis similar to that of System SQ0 , we have: u(β(N )µ0 ) π SQ (N ), (1 − u(β(N )µ0 ))2 E[LSQ ] = E[QSQ ] + ρ(β(N )µ0 ).

E[QSQ ] =

(9) (10)

12

And the stationary probability of N customers in service π SQ (N ) is: N −1 X

π SQ (0) =

j=0

π

SQ

(k) =

ρj (β(N )µ0 ) ρN (β(N )µ0 ) + j! N !(1 − u(β(N )µ0 ))

( k 0) π SQ (0) ρ (β(k)µ k! π

SQ

!−1 ,

0 ≤ k ≤ N,

N )µ0 ) k−N (0) ρ (β(N u (β(N )µ0 ) N!

k > N.

Therefore, the expected waiting time and the total time in the SQ system are: 1 N ρN +1 (β(N )µ0 ) E[QSQ ] , = π SQ (0) λ λ N !(N − ρ(β(N )µ0 ))2 E[LSQ ] 1 E[T SQ ] = + E[W SQ ] + t(N ). = λ β(N )µ0

E[W SQ ] =

(11) (12)

3.5. Parallel Queue system with workload dependent service rate: System P Q Let π P Q (L) be the stationary distribution of LP Q . From the balance equations for one server in the P Q system, we obtain the distribution for the number of customers in an individual server’s line: −1


λ k

∞ X  π P Q (0) =  1 + 

N k Q

k=1

sP Q (l)

  

,

l=1

π

PQ

(k) = π

PQ

(0)


λ k N k Q

.

sP Q (l)

l=1

Let γ capture the increase in expected service rate due to workload-dependent speedup: γ = ∞ P E[sP Q (L)] ≥ 1, where E[sP Q (L)] = π P Q (l)sP Q (l). µ0 l=1

PQ

Lemma 2. E[s

(L)] =

λ N (1−π P Q (0))

and hence, γ =

λ . N µ0 (1−π P Q (0))

When the number of customers in a queueing system is 0 or 1, the queue size is 0. Let π ˆ P Q (k) be the stationary distribution of the queue size in System P Q:  PQ π (0) + π P Q (1) if k = 0, PQ π ˆ (k) = π P Q (k + 1) if k ≥ 1. Then, the expected number of customers in queue and in the system are:
∞ ∞ λ k+1 X X PQ PQ PQ N E[Q ] = lˆ π (l) = π (0) k k+1 , Q PQ l=0 k=1 s (l)

(13)

l=1

E[L

PQ

]=

∞ X l=0

lπ

PQ

(l) = π

PQ

(0)

∞ X k=1

k


λ k N k Q l=1

sP Q (l)

.

(14)

13

From Little’s Law, we obtain:

E[W

PQ

]=

E[QP Q ] λ N

=π

PQ

(0)

∞ X k=1

k k+1 Q


λ k N

,

(15)

sP Q (l)

l=1 PQ

E[T P Q ] =

E[L

λ N

]

= E[W P Q ] +

1 1 . = E[W P Q ] + E[sP Q (L)] γµ0

(16)

4. Comparative analysis In this section, we perform four separate comparisons and discuss the findings associated with each. The comparisons are: 1) System SQ0 to System SQ, 2) System P Q0 to System P Q, 3) System SQ0 to System P Q0 , and 4) System SQ to System P Q. The comparisons are clearly illustrated in Figure 2.

System'SQ0$

Single'Queue!with!! fixed!service!rate!

Sec5on!4.2!

Sec5on!4.1!

System'SQ$

Figure 2

System'PQ0$

Parallel'Queue!with!! fixed!service!rate!

Sec5on!4.1!

Sec5on!4.3!

Compare!SQ!to!PQ!according!to:! Single'Queue!with!server! 4.3.2!–!Expected!Wai5ng!Time! slowdown!due!to!social!loafing! 4.3.3!–!Expected!PreEService!Time! 4.3.4!–!Expected!Total!Time! ! ! Comparison framework and map of results !

System'PQ$

Parallel'Queue!with!server! speedup!due!to!conges5on!

In subsection 4.1, we focus on isolating the effects of each behavioral factor. Namely, we compare the P Q and SQ systems to their respective benchmarks P Q0 and SQ0 to analyze the impact of speedup in a P Q system and the impact of social loafing in a SQ system. In subsection 4.2, we focus on the effect of the walking time by comparing the benchmark systems P Q0 and SQ0 to each other. In subsection 4.3, we combine all three factors by comparing the P Q and SQ systems to each other according to three different performance measures, deriving performance thresholds that can be used to guide managers in making queueing system design decisions in real-world service operations settings. To illustrate our findings, we complement our analytical results with numerical examples.

14

4.1. Impact of behavioral factors on system performance We compare the P Q and SQ systems to their respective benchmarks P Q0 and SQ0 to analyze the impact of workload dependent speedup in a P Q system and the impact of social loafing in an SQ system. First we compare the distributions of queue size and number of customers in the system stochastically. Proposition 1. In the P Q system, the benefit of speedup on queue length and number of customers in the system is stochastically greater than that of a comparable average increase in expected service rate (i.e., increase from µ0 to γµ0 in the P Q0 system): LP Q 0) Figure 3

In shaded (non shaded) area, the waiting time under SQ system is lower (higher) than under P Q system.

18

The numerical example in Figure 3 illustrates how the waiting time threshold value determines the choice of structure between P Q and SQ, and how this decision can change based on the number of servers in the system. The shaded (non-shaded) areas in the figure represent the regions where SQ results in the shortest (longest) expected waiting time. In Figure 3(a), we set θS = 0 to focus only on the effect of social loafing. The other parameters are set as follows: µ0 = 2, u ˆ = 0.9, θL = 0.05, and KL = 4. For a set utilization value, the impact of pooling is increasing, then decreasing in N . The impact of social loafing is, however, increasing in N at a decreasing rate, which explains the interesting non-monotone decision rule for selecting the queue design configuration. When there is no speedup effect, the SQ system dominates the P Q system when the number of servers, N , is between 2 and 12 in our numerical example. The P Q system dominates in all other cases. In Figure 3(b), we add the workload dependent speedup effect (namely, θS = 0.02, MS = 20, and KS = 4 ) and demonstrate how the shaded area (i.e., the area where SQ dominates P Q) shrinks as a direct result of the speedup effect. When the speedup effect grows larger, the region where the SQ system dominates the P Q system becomes smaller. Our numerical results are consistent with what is typically observed in real-world situations: SQ is common for small queueing environments (e.g., drugstores with 3-5 registers), but not as common for large queueing systems (e.g., Target or Wal-mart stores). Based on the results shown in Figure 3(b), if the speedup effect increases the service rate by more than 4%, the P Q system always outperforms the SQ system. It is very interesting to note that α > γ (as can be seen in Figure 3(b)), which clearly illustrates that the overall performance benefits associated with the speedup effect are larger than those obtained by a comparable average increase in expected service rate. When social loafing is also considered in the SQ system, the speedup effect in the P Q system enlarges the dominance region of P Q systems over SQ systems. The numerical example presented in Figure 4 illustrates how the performance threshold value for expected waiting time, αw , increases as the utilization parameter u ˆ is reduced from 0.9 to 0.8. Based on Figure 4(a), notice that at the 0.85 and 0.8 utilization levels, the expected waiting time threshold curve is always above the level α = 1, indicating that, without the workload-dependent effect, the P Q system is never preferred. Also note that the speedup effect, α, is reduced as the utilization level decreases: with speedup effect of θS = 0.02, α = 1.0299 for u ˆ = 0.8 and α = 1.032 for u ˆ = 0.85, which is still insufficient or the P Q system to become preferable. These results suggest that, for lower utilization values, the impact of social loafing in the SQ system has to be quite pronounced in order for the P Q system to become more attractive. We demonstrate this finding in Figure 4(b): as we decrease the parameter KL (KL = 2.3) which controls the degree of the social loafing effect (lower KL indicates a higher social loafing effect), the dashed curve corresponding to

19

the utilization of 0.85 recedes below the α = 1 level when the number of servers exceeds 10, making

Threshold value for expected waiting time

the P Q system preferable.

2 u = 0.8 u = 0.85 u = 0.9

1.8

1.6

1.4

1.2

1

0.8 0

2

4

6

8

Number of servers (N)

10

12

14

(a) Low impact of Social Loafing: KL = 4

Threshold value for expected waiting time

2

u=0.8 u=0.85 u=0.9

1.8

1.6

1.4

1.2

1

0.8 0

2

4

6

8

Number of servers (N)

10

12

14

(b) High impact of Social Loafing: KL = 2.3 Figure 4

The threshold αw decreases in utilization: at lower utilization values, a very large social loafing effect is needed to make SQ system perform worse than the P Q system.

20

4.3.3. Expected pre-service time In this subsection, we focus on selecting the queueing system structure (SQ or P Q) based on expected pre-service time. The pre-service time captures the time between when a customer enters the system and when she starts receiving service. In our model, this time includes the time that the customer spends waiting in queue and walking from the head of the SQ to the server. Earlier, in Lemma 1, we established a threshold on walking time that guides the system choice when the impacts of speedup and social loafing are negligible. We next derive the pre-service performance threshold value for α, denoted as αp , such that for α ≥ αp , the P Q system will outperform the SQ system in terms of expected pre-service time. ! r  −1 N +1 ρ ρ0 , Proposition 3. There exists αp = 2N 1 + 1 + 4N π SQ (0) (N −1)!β(N )N0−1 (N β(N )−ρ0 )2 + λt(N ) such that 1. E[W SQ ] + t(N ) ≥ E[W P Q ] for all α ≥ αp , and 2. αp < αw . Proposition 3 provides a decision rule that can guide the choice between a P Q or SQ system, based on the minimum expected pre-service time. We present numerical examples to illustrate this decision rule and provide valuable insights about the threshold αp . Figure 5 illustrates the optimal choice of queue structure (SQ or P Q) based on the expected pre-service time. We use the same numerical values in Figure 5 as in Figure 3(a) to demonstrate how the shaded region (the region where the SQ structure is optimal) shrinks when walking time is considered. Recall that a longer walking time makes the SQ system less attractive as customers spend additional time walking from the head of the queue to the server. In this example, we set the walking time between two adjacent servers at τ = 0.5. Figure 5 shows that when speedup effects are negligible, the SQ system outperforms the P Q system when the number of servers is between 3 and 9, and the P Q system dominates for all other values of N . As in Section 4.3.2, when the speedup effect intensifies, the α line shifts up, thereby shrinking the region where SQ dominates. In Figure 6, we investigate the sensitivity of the pre-service time performance threshold, αp , with respect to both utilization (Figure 6(b)) and walking time between two adjacent servers τ (Figure 6(a)). Based on Figure 6(b), we see that the threshold αp is highly sensitive to utilization. For example, when N = 5, and the utilization is high (ˆ u = 0.9), the threshold value is relatively small (αp = 1.018). However, when the utilization drops, the threshold values are much larger (αp = 1.093 u ˆ = 0.85, and αp = 1.118 for u ˆ = 0.8). It is also interesting to note that the threshold αp is not well-ordered in utilization (as shown in Figures 4(a) and 4(b)). In our example, when the number of servers, N , is between 2 and 7, αp is higher for u ˆ = 0.8 than it is for u ˆ = 0.85, but lower when N is larger or equal to 8. As the utilization drops, the impact of walking time on system performance becomes more pronounced for the SQ system, particularly in larger systems. From Lemma 1,

21

Threshold value for expected pre−service time

1.03

PQ

SQ

PQ

1.02

αp x

1.01

α

1

0.99 0.98 0.97 0.96 0

Figure 5

2

4

6

8

Number of servers (N)

10

12

14

In shaded (non shaded) area, the waiting time under SQ is lower (higher) than under PQ

the expected t(N ) is independent of the utilization and increases in a fairly linear manner in the number of servers N . As the utilization decreases, the expected time in queue is reduced, and thus, the expected walking time t(N ) winds up representing a larger portion of the expected pre-service time, making the SQ structure less attractive. When we compare αp to α for different values of utilization u ˆ in Figure 6(b), we see that the SQ dominance regions are not nested: the SQ system is preferable when the number of servers is between 3 and 15 for an utilization level of u ˆ = 0.85, but the dominance region shifts to a number of servers between 2 and 11 when u ˆ = 0.8. In sum, the choice of an optimal queueing structure is highly sensitive to both the utilization and the size of the system. This is an important observation in practice, as fluctuations in demand are inherent to real-world situations (due to seasonal demand, promotion, competition etc.). As such, managers should carefully select the design and the size of their queueing operations (represented by the number of servers in our model) to be robust for a wide range of system utilization levels. 4.3.4. Expected total time in system Finally, we look at the choice of the optimal queue system (P Q or SQ) from the perspective of the total time (i.e., pre-service time plus service time). Let δ be such that: E[T P Q0 (δµ0 )] = E[T P Q ].

(26)

This means that the P Q0 system with service rate δµ0 , which is independent of the workload and number of servers, performs as well as the P Q system with respect to total time in system. Lemma 5. There exists a unique δ > γ > 1 that solves equation 26: δ = u0 +

γ Aγµ0 + 1

(27)

22

Threshold value for expected pre−service time

1.05 = 0.2 1.04

= 0.3 = 0.4

1.03

= 0.5

1.02 1.01 1 0.99 0.98 0.97 0.96 0

2

4

6

8

10

12

14

Number of servers (N) (a) Sensitivity to walking time between adjacent servers τ .

Threshold value for expected pre−service time

1.12 1.1

u = 0.8 u = 0.85 u = 0.9

αp

1.08 1.06 1.04

α=1.032

1.02

α=1.029

αp

1 α=1

0.98 0.96 0

2

4

6

8

Number of servers (N)

10

12

14

(b) Sensitivity to utilization u. Figure 6

Sensitivity of the threshold value αp with respect to u and walking time between adjacent servers τ .

Next, we derive the threshold value for δ, denoted as δT such that the P Q system will outperform the SQ system in terms of total time in the system when δ > δT .

23

Proposition 4. There exists δT =



1 β(N )

π SQ (0)N ρN +1

0 + λ1 N !β(N )N −1 (N β(N µ + t(N )µ0 )−ρ0 )2 0

−1

+ uo , such that

E[T SQ ] ≥ E[T P Q ] for all δ ≥ δT . Proposition 4 provides a decision rule that can be used to select the P Q or SQ system based on the expected total time. The difference between αp and δT is only in expected service time and hence, is very small, especially when the utilization is large (as illustrated in Figure 7(b) with u ˆ = 0.9). Therefore, most of the intuition discussed for the pre-service threshold αp applies here as well. In this case, the expected service time accounts for only a small portion of the expected total time. However, when the utilization is low (e.g. u ˆ = 0.3 in Figure 7(a)), the difference between the two thresholds may become noticeable since the expected wait time in queue is low and consequently, the expected service time represents a relatively larger portion of the total time in the system. Hence, as illustrated in Figure 7, the system choice based on the expected pre-service time prescribed in Section 4.3.3 may not be optimal to guarantee the best total time in the system.

5. Conclusion While traditional queueing theory clearly establishes the superiority in operational performance of SQ systems, many managers in service organizations have been reluctant to combine queues in practice. To explain this discrepancy, we examine the effects of three commonly observed behavioral and physical layout factors that can potentially undermine the attractiveness of the SQ system, when compared to its P Q counterpart. Specifically, we investigate the effects of (1) social loafing, which decreases servers’ rate in the multi-server SQ system, (2) workload-dependent speedup, which increases service rate in the multi-server P Q system, and (3) walking time from the head of the queue to the server, which further diminishes the efficiency of the SQ system. To the best of our knowledge, our work is the first to incorporate these factors into queueing models and to analyze their impact on the performance of queueing systems. We show that in the P Q system, the workload-dependent speedup experienced by servers reduces queue length, resulting in lower expected wait time, and total time in the system than a comparable increase in expected service rate would produce. We demonstrate that the aggregate benefits associated with the workload-dependent speedup effect consist of both the direct benefits resulting from a comparable increase in expected service rate and the indirect benefits that the smoothing of each individual queue brings to the entire system. Ultimately, the choice between the SQ and P Q system depends on the magnitude of the three effects mentioned above and their impact on the operational performance of the system. We propose functional forms for the three effects, and we hold that the parameters of these functional forms, which capture the magnitude of these effects,

24

Threshold value for expected total time

1.5

αp x

1.4

δT

1.3 1.2 α=1.0158

1.1 1

δ=1.0111 0.9 0.8 0

2

4

6

8

Number of servers (N)

10

12

14

(a) Low utilization: u = 0.3.

Threshold value for expected total time

1.05

αp α=1.035

x

1.04 1.03

δT

δ=1.0349

1.02 1.01 1 0.99 0.98 0

2

4

6

8

Number of servers (N)

10

12

14

(b) High utilization: u = 0.9. Figure 7

When utilization is low, system choice according to minimum expected pre-service time may be different from the system choice according to the minimum expected total time; When utilization is high, the difference between αp and δT is very low and hence, the system choice according to either measure is similar.

25

can be estimated empirically (e.g., using data fitting) for any queueing system. We derive threshold values for the aggregate effect of speedup according to three different performance measures: (1) expected waiting time, (2) expected pre-service time, and (3) expected total time. These parameter estimates taken together with the threshold values can assist managers in their system selection process (as shown in Section 4). By complementing our theoretical results with numerical examples, we obtain the following insights. (1) For a moderate speedup effect and high utilization, the optimal choice between a P Q and SQ system, for all three performance measures, depends on the number of servers in the system in a non-monotone way. That is, the P Q system dominates the SQ system when the number of servers is either small or large enough, but not in-between. As the speedup effect becomes more pronounced, the region over which the P Q system dominates the SQ system becomes larger. (2) For expected pre-service time and total time in system, the region where the SQ system dominates the P Q system is not nested and very sensitive to the utilization of the system. These findings make it clear that the size of the queuing system is a critical determinant of performance. Consequently, the robustness of both queueing system size and design decisions should be carefully considered under varying levels of demand rate. In this work, we highlight the importance of accurately designing service systems as customer wait time has been shown to affect the financial performance of the firm (Allon et al. 2011). We believe that our theoretical formulations, numerical analyses, and insights will not only provide managers with guidance in the design of their queueing operations, but also serve as building blocks for researchers in their future work on behavioral queueing.

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6. Appendix Proof of Lemma 1

We assume that the end of the pooled (single) queue is at the middle of the n P n(n+1) kτ = 2T = τ n(n+1) ; When server range. When N = 2n + 1, the expected walking time is: N2 N 2 2n+1 k=1 n−1 n−1

τ  n(n−1) n  τ n2 P P 1 N = 2n, the expected walking time is: N2 k + 12 τ = 2τ k + =n + 2 = n 2 = τ n2 . N 2 2 k=0

k=0

Proof of Lemma 2 π

PQ

(l) = π

PQ

(0)


λ l N l Q

sP Q (k)

k=1

PQ

E[s

∞ ∞ X X 1 1 PQ (L)|L ≥ 1] = π P Q (l)sP Q (l) = π (0) sP Q (l) l Q P r(L ≥ 1) l=1 1 − π P Q (0) l=1 k=1




λ l−1

∞ X  1 λ  PQ PQ N 1 +  π (0) s (l)  l Q 1 − π P Q (0) N P Q l=2 s (k) k=1  
l ∞ λ X  1 λ  PQ PQ N 1 +  = π (0) s (l)  P Q l Q PQ  1 − π (0) N l=1 s (k)

=

k=1


λ l N sP Q (k)

28

=

∞ λ λ X PQ 1 N π (l) = 1 − π P Q (0) N l=0 1 − π P Q (0)

This result is restating Little’s Law since 1 − π P Q (0) is the probability of busy server and 1 − π P Q (0) =

λ N

E[sP Q (L)|L≥1]

. From here, we see that γ =

Proof of Proposition 1

λ . N µ0 (1−π P Q (0))

We provide the proofs for LP Q m, such that:  PQ π (l) ≥ π P Q0 (γµ0 , l) if l ≤ m, ˆ π P Q (l) ≤ π P Q0 (γµ0 , l) otherwise. Next, we show by contradiction that K X l=0

π P Q (l) ≥

K X l=0

π P Q0 (γµ0 , l) ∀ K ∈ Z+

(31)

29 ˆ ˆ K K P P + PQ ˆ Suppose not, it means that there exists some K ∈ Z such that π (l) < π P Q0 (γµ0 , l). l=0

This implies: 1 =

∞ P

π P Q (l) =

l=0 ∞ P

ˆ K P

π P Q (l) +

∞ P

π P Q (l)