Signaled Queueing. Laura Brink, Robert Shorten, Jia Yuan Yu ABSTRACT. Categories and Subject Descriptors. General Terms. Keywords

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1 Signaled Queueing Laura Brink, Robert Shorten, Jia Yuan Yu ABSTRACT Burstiness in queues where customers arrive indeendently leads to rush eriods when wait times are long. We roose a simle signaling scheme to decrease wait times by distributing customer arrivals more uniformly. Agents receive one of several signals with suggestions on what time to join the queue. We quantify the efficiency gains, both analytically and emirically, with resect to a number of arameters of the roosed signaled queue, such as burstiness of arrivals, number of distinct signals, and roensity of customers to follow suggestions. Categories and Subject Descritors I.. [Distributed Artificial Intelligence]: Multiagent systems General Terms Management, Design, Performance eywords Queueing, Signaling, Social welfare. INTRODUCTION Queueing roblems are encountered throughout multiagent systems, oerations research, economics, management, and telecommunications. Many queueing models have been roosed and analyzed in the literature. Some areas that have been examined so far are the equilibrium attern of customer and oerator s interactions [3], the attern that customers abandon queues based on wait times [5], and alying games models to dynamic queueing roblems [3]. While the literature on queueing theory is very large, the use of signals to decrease wait times aears to be under exlored. Our starting oint is the observation that while queues are found in many laces in every-day life, not all queues should behave in the same way. Most queues oerate according to a simle FIFO rincile. Customers that arrive first are served first. Examle of this tye of queueing can be found at check-in in airorts, in the grocery store, and in the doctor s surgery. In each case, there exists a scoe for alying Aears in: Proceedings of the 4th International Conference on Autonomous Agents and Multiagent Systems AA- MAS 5), Bordini, Elkind, Weiss, Yolum eds.), May, 4 8, 5, Istanbul, Turkey. Coyright c 5, International Foundation for Autonomous Agents and Multiagent Systems All rights reserved. signaling to imrove the queue erformance. First, in some situations, the FIFO first-in first-out) aradigm is not otimal. Although oerators have more information than customers, they do not take advantage of it, e.g., airlines know when each ucoming flight will leave. In other situations, the use of FIFO queues is simly not aroriate. For examle, in the doctor s surgery, one should rioritize atients according need or likelihood of infecting other atients, or of being infected by others). Such rioritization can be imlemented via a signaling scheme. Finally, even in the FIFO queue, bursty arrivals can lead to oor queue erformance. For examle, at airorts, customers arrive in bursts, i.e., the arrival rate fluctuates highly over time. These bursts cause long wait times because many eole are entering the queue at the same time. A signaling system can also be used at an airort to make wait times shorter. When customers check-in online for their flights, they receive a signal suggesting them what time to arrive at the queue, along with a conditional reward as incentive to comly. The signals will redistribute customers arrival times such that arrivals are saced out and arrival bursts do not occur. This aroach can be generalized to any system that has a method to deliver signals to the customers, flexibility in customer arrival times, and bursts of customer arrivals. To encourage customers to act as intended by the signals, incentives based on the difference between their signals and their actual arrival times can be used. In this aer, we start from a multiagent FIFO queueing system Section 3). We then augment the FIFO queue with a signaling scheme and a corresonding model of customerresonse Section 4). The oerator sends a distinct signal to each customer. Each signal is simly a set of suggested arrival times. This is done only once before the arrival of the first customer. In turn, each customer can slightly modify or modulate its arrival time according to the received signal. To make our analysis tractable, we make an imortant behavioral assumtion on how customers modulate their arrival times in resonse to the signals received. The modulated arrival time is a convex combination of the nominal arrival time when there is no signaling) and the closest suggested time in the customer s signal set. Although this behavior is simlistic, it does reflect the reality of human resonse to signals [4] and can be modified accordingly to include more comlex behaviors). By its linearity, it is even more reflective of machine resonse. Our roosed signaling scheme works in many queueing settings, as long as these settings share the following characteristics. A roducer of a resource e.g., a commodity or

2 service), a number of consumers. First, the consumers communicate an intent to consume the resource e.g., reayment, registration), the roducer communicates signals to the consumers along with incentives for following these signals, the consumers choose when to consume the resource We assume that customers have some flexibility as to when they arrive. Our signaling solution is esecially useful when the environment is raidly changing, i.e., where agents do not have sufficient time to learn or adat their arrival times to those of other agents. Moreover, just as signaling reduces wait times in queues, it is easy to see that signaling can also reduce eak-demand in ower networks and congestion in road networks. Through a combination of robabilistic analysis Section 4.) and simulation Section 5), we quantify the erformance imrovement of the signaled queue as a function of a number of arameters of the signaling scheme, of the arrival attern of customers, and of the resonsiveness of customers to the signals. In many cases, the erformance imrovement is significant. Examle Airort). Our signaling solution can be alied to an airort check-in setting. Uon booking a flight, each customer receives an additional signal. This signal indicates a number of suggested arrival time at check-in. For examle, a customer may receive the signal 5 and have the otion of checking out at :5am, :5am, etc. When the customer arrives at check-in, he receives a reward based on how close he arrived to one of the suggested times of his signal. The customer could also receive an extra reward based on the length of the queue at arrival. The rewards can be monetary or otherwise.. RELATED WORS Our work builds uon queueing systems and multi-agent systems. In articular, the notion of wait times that we take as our erformance metric has been characterized in the case of FIFO queues in []. Alternative modes of riority to arrival time in a queue have been roosed in numerous works e.g., [9]). These modes include: agent-riority [], shortest-job first [], and multile queues [6]. All of these do not exhibit signaling between an oerator and the customers. The roblem of otimization of multile queues or queueing networks is usually modeled as a Markov decision roblem [8]. In contrast to our work, these lines of work do not exhibit signals sent by the oerator as control variables. Our signaling aroach is an alternative to other existing aroaches to managing queue, such as advanced booking, reservations, and differential ricing deloyed at theme arks e.g., Disney arks, Sea World). Otimization of a single queue has been studied when the service rate µ in our notation) or the arrival rate λ i in our notation) of customers can be controled [7]. This is however not the case in our model, where the oerator only controls the signals sent to customers. So-called Active Queue Management algorithms [] have been roosed to deal with burstiness in communication networks by droing ackets in real-time. However, such aroaches are useless when we exclude the otion of turning customers away. A otential feature of our roosed solution is that it not emloy any real-time control action: all signals are generated and sent at a single time instant. This feature also sets our work aart from the studies of queueing systems with feedback []. In our work, we adot a model of a heterogeneous oulation of customers characterized by samles from a robability distribution, which is commonly used in statistics to model aggregate effects [8]. The intricate behavioural asects of queueing customers, although widely studied [5, ], is not a focus in this aer. The works most closely related to ours concern multiagent queues, where the agents can interact strategically as in the case of queueing games [5]), or by following fixed olicies as in our model. In articular, two notions of signals have been introduced in such queues. [3] gives an overview of game theoretic asects of queueing among utility-maximizing agent. In contrast to the notions of signaling of [3] where the signal rocess is tyically the length of the queue at every time instant, we consider a set of signals that are sent only once at the start of time. [5] examines how wait time announcements affect the actions of customers in a multi-server system. These announcements are udated in real-time, whereas the signals in our roosed system are sent once before the first customer arrival. Beyond obvious alications in managing queues of human customers in cities [4], our solution aroach can be alied in a number of areas, notably communication network queues [7], transortation network queues [9]. The queueing liteature is extensive and our survey is not exaustive. However, this is one of the first times that signaling and queues have been combined to regulate arrival rocesses. 3. FIRST-IN FIRST-OUT QUEUES Most queues oerate on a first come, first served basis. In a first-in first-out FIFO) queue, the erson who comes in earlier is served earlier. In such situations, burstiness can cause the queueing system to be overwhelmed, and to excessively long wait times. We examine a simle FIFO system; there are i {,..., N} customers and only one server. The service rate is µ, where /µ is the mean service time. For simlicity of resentation, we will examine a secial case of the M/M/ queue [6] where the service time is the same for each customer, i.e., a constant service time a =. Note that these assumtions µ are to aid exosition and that our results generalize to i.i.d. service times in a straightforward fashion. Each customer arrives at the end of the queue at a nominal arrival time x i. We will assume that x x x 3... x N. Let e, e,... denote indeendent exonential random variables with arameters λ, λ,..., resectively. As in the case of M/M/ queues, we assume the time between consecutive arrivals are indeendent and exonentially distributed, such that i x i = e j, for all i =,..., N. j= The arameter λ i is the arrival rate of the i-th customer; a large value of λ i indicates that more customers are arriving er unit time. Although µ is a constant, the arrival rate λ i varies over the the customer index i and hence over time. This varying arrival rate models arrival bursts, when many customers join the queue in a short time eriod, causing long wait times. Each customer leaves the queue after some waiting eriod, or nominal wait time, d i. The time that each customer leaves the queue is then x i + d i. The wait time of each customer

3 ... R Figure : Illustration of sequences of suggested times corresonding to different signals. There is a shift of / between sequences corresonding to consecutive signals, and a eriod of between suggested times within each sequence. deends on the wait times of the customers in front of him. In general, every other customer after the first customer has a random) wait time of d i = max{a + x i + d i x i, }. The traditional FIFO queue oerator has no control over the wait times {d i} due to the lack of actuation caabilities. Our objective is now to reduce the total wait times of all customers by introducing a new queueing system that involves signals sent by the oerators to the customers before they arrive. 4. NEW QUEUEING SYSTEM WITH SIG- NALS We aim to decrease wait times by having the oerator send signals to all of the customers incentivising them to arrive at suggested times. We call the resulting system a signaled queueing system or signaled queue. For examle, when airline customers check-in online, they may be offered a discount on future flights, if they arrive at the check-in desk at a certain time. At time t =, the oerator icks two arameters: a real number > called the eriod, and an integer called the number of signals. The oerator samles N i.i.d. random variables s,..., s N from an uniform robability distribution over suort {,,..., }. Then, for every customer i =,..., N, the oerator sends the signal s i. The value of the signal corresonds to one of the following sequences of suggested arrival times:,,,..., if s i = δ, + δ, + δ,..., if s i =. )δ, + )δ,..., if s i =, where for simlicity, we write δ = / to denote the interval between consecutive signal values. The suggested arrival times corresonding to different signals are also illustrated in Figure. 4. Customer resonse In our roosed queueing system, each customer s new or adjusted arrival time at the end of the queue, y i, will be a function of their original arrival time or the nominal arrival time, x i, the signal they received, s i, and a arameter σ i that models how much customer i ays attention to the signal: y i = fx i, s i, σ i). We now roose a model for how customers will resond to the signals. The arameter σ i reresents customer i s tendency to follow the signal, i.e., arrive at a time close to a suggested arrival time. To model a heterogeneous oulation of customers, we assume that σ,..., σ N are i.i.d. random variables with the interval [, ] as suort. When σ i =, the customer ays no attention to the signal and behaves as if he had never received a signal, and therefore, y i = x i. When σ i =, the customer arrives at a suggested arrival time close to its nominal arrival time x i. For simlicity, we take this suggested arrival time as x i + s i ), which is either the closest suggested arrival time below or above the nominal arrival time x i. In sum, we consider the following instance of this function y i = fx i, s i, σ i) as a model of customer resonse: ) y i = σ i)x i + σ i xi + si, for i =,..., N, where denotes the round-down-to-the-nearest-integer oerator. The equation satisfies our constraints that y i = x i when σ i = and y i = x i + s i ) when σ =. The set of y i is not necessarily sorted in ascending order. Let y ) y )... y N) denote the sequence of sorted adjusted arrival times. We define the corresonding adjusted wait times h,..., h N. Assuming that h =, for every i =, 3,..., N, the wait time for the customer arriving at time y i) is h i = max{a + y i ) + h i y i), }. 4. Performance Comarison In this section, we comare the erformances of the signaled system and the FIFO system by examining the wait times exerienced by customers. Our objective is to show that these wait times are shorter in the signaled case. The following theorem gives a erformance guarantee on signaled queues, relative to FIFO queues. It says that we exect that the wait-time exerienced by customers in the FIFO queue to be larger than in the signaled queue. In articular, the theorem bounds the robabilistic erformance imrovement. To set notation, recall that h i is the wait-time of the i-th customer to arrive in the signaled queue, whereas d i is the wait-time of the i-th customer to arrive in the FIFO queue. Remark. The customers corresonding to d i and h i need not the same hysical customer. Remark Comaring d i and h i). How meaningful is a comarison between d i and h i? Arguably, customers can comly with the suggested arrival times and wait outside the queue before joining the queue. In this case, such customers exerience two wait times: inside the queue and outside the queue. We argue that the wait time outside the queue can be ut to better use than time in the queue: having a coffee or shoing in the airort check-in examle. Theorem. Let µ = Eσ i and µ = Es i/. Let v = Vσ is i/) = Eσi Es i/) Eσ i) Es i/). For every customer i {,..., N}, every ζ >, every z, and

4 FIFO With signals \zeta Figure : Plots of Pd i > ζ + z d i = z) in solid line and the bound on Ph i > ζ + z h i = z) in broken line versus ζ, as derived in Theorem. We used the following set of values of the arameters: a = 5, = 5, λ = 4, ζ [, a], γ such that the three fractions on the right-hand side of ) are aroximately equal. every γ, we have: Pd i > ζ + z d i = z) = e λa ζ), ) and Ph i > ζ + z h i = z) + Nλ γ N/λ) + Nλ ζ a γ)/ N/λ) v ζ a γ µ. ) µ ) The roof of Theorem aears in the Aendix. The robabilities of Theorem are further illustrated in Figure. We observe that indeed for all values of ζ, the uer bound on Ph i > ζ + z h i = z) is less than Pd i > ζ + z d i = z). Theorem shows that customers in the signaled queues are less likely to wait for a long time than customers in the FIFO queues. As further illustration, Figure 3 shows emirically that the signaled system s wait times are more likely to be small than the FIFO system s wait times. 5. EMPIRICAL RESULTS This section resents in-deth simulations to illustrate the erformance of signaled queues when comared to FIFO queues. Thus, this section comlements emirically the revious section. Secifically, we examine how the values of different arameters affect wait times in the signaled queueing system. Questions to be examined include: how many distinct signals there should be; what should the eriod be; and finally, how much should customers adhere to their suggested times for the system to work adequately. To this end, we define the relative erformance of the signaled queue comared to the traditional FIFO queue as the ratio between the sums of adjusted and nominal wait times: Ni= h i Nj= d j. If this ratio is significantly smaller than one, then signaling is indeed decreasing the aggregated customers wait times in a noticeable manner. Throughout our simulations, we set the number of customers N to 5, the service time a to 5. To model bursty arrivals, the arrival rate sequence {λ i} is iecewise-constant, alternating between a high value and a low value every N/4 consecutive customers cf. to of Figure 8). Each simulation scenario is reeated over trials, and average relative er Histrogram Comaring Nominal and Adjusted Delays Nominal Delays d) Adjusted Delays h) Figure 3: Histogram comaring the robability distributions of the sum of FIFO wait times, N N j= j= di, and the sum of adjusted wait times, hi, for, trials and one hundred customers, N =. There is a higher robability that the sum of the adjusted wait times will be within the first bin than the sum of FIFO wait times to be within the first bin. formance is lotted along with error-bars corresonding to one standard deviation. 5. Relative Performance versus δ The oerator can vary the length of time between two consecutive signals, the interval δ = /. We set σ i = for every i all customers arrive at the nearest suggested time). First, we investigate the total time y N) y ) in the signaled queue comared to the total time x N x without signals. For the case δ = a, Figure 4 shows the ratio y N) y ) x N x as a function of the number of distinct signals. As exected, this ratio is close to for small values of. However, as increases, so does the eriod = δ, which leads to large values of x N + s N ). Figure 5 comares the relative erformances for three values of δ. A larger value of δ leads to a better erformance. This is exected because the more time between customer arrivals, the lower their wait times. However, a large δ also leads to a large eriod and a large total time y N) y ). The number of signals er eriod,, affects the difference between a customer s nominal arrival time and the nearest of his suggested arrival times. This means that the customer would have to make a large adjustment to follow the suggestion. Figure 6 shows that when is large, the average relative adjustment required from the customer can become non-negligible. 5. Relative Performance versus Burstiness We now investigate the relative erformance for different burst atterns, as modeled by the sequence {λ i}. During a burst, many customers arrive in a short amount of time, according to a high arrival rate, λ. We consider the burst atterns of Figure 7. In each attern, there is the same

5 Percent Difference of Total Time N=5,, sigma=, delta=a Relative Performance N=5,, sigma= delta=a delta=.5a delta=.5a 5 5 Number of Different Signals ) Number of Different Signals ) Figure 4: The relative difference between the nominal total time and the adjusted total time, y N) y ) x N x, deends on the number of signals. As increases, the signaled system takes more time than the FIFO system. number of customers with a high arrival rate, who arrive in one or multile bursts. Figure 8 shows that the relative erformance imroves as bursts rush eriods) become shorter and more numerous). Finally, in Figure 9, we examine how the relative erformance deends on the value of the arrival rate λ during bursts, relative to the normal one λ. The larger the difference between the two, the better is the relative erformance of the signaled queue. 5.3 Relative Performance versus σ i In the revious simulations, the distribution of the random variable, σ i, that determines how carefully each customer follows its suggested arrival times, has been ignored we have set σ i = uniformly. We had assumed that all customers follow their suggested arrival times exactly. In this section, we take {σ i} as i.i.d. samles from a robability distribution over the suort [, ]. To make things more realistic, we choose the beta distribution with shae arameters α and β cf. Figure ), and corresonding robability density function gx) = Bα, β) xα x) β, x [, ]. where the B denotes the beta function. Figure shows how the shae arameters of the beta distribution affect the relative erformance. In Figure, we comare the different beta distributions with the ideal case, σ =, and the worst case, σ =. When σ =, all of the customers follow their signals with some error. When σ =, none of the customers follow their signals and they all arrive at their original arrival times, thus the system erformance is always because none of their arrival times change. When σ is determined by a beta distribution, there are different robabilities for the degree a customer listens to the suggested arrival times. The robability den- Figure 5: Comaring the relative erformances of different ratios between the interval and service time when varies from to,. When δ < a, customers are arriving faster than customers are leaving so the erformance of the system is worse. When δ > a, customers are arriving slower than the rate that customers are being served so the erformance is good because not many eole are waiting. However, when δ > a, the total time for the signaled system is much larger than the FIFO system. We choose δ = a because it saces out the signals enough such that the system erformance is good but not too much that the total time is too large. sity functions of the beta distributions used in Figure are shown in Figure. Figure shows that the smaller the shae arameter β, the better the system erformance. When the mean of the beta distribution tends towards one, the system erformance increases because the robability that customers listen to their signals increases. The more distribution weight is near, the better the signaled queue erforms. One way to ensure customers act as intended by the signals is to give incentives based on the difference between their signals and their actual arrival times. Remark 3 Incentives). Rewards may or may not be necessary to ensure that customers listen to their signals. If the oerator chooses to use rewards, these do not have to be monetary. For examle, the crowd-sourcing traffic a Waze gives customers incentives in the form of oints when they reort traffic incidents. The underlying human resonse to incentives is however beyond the scoe of this aer. APPENDIX A. PROOFS Proof of Theorem. The roof is broken down into four arts. First, we derive the robability that the FIFO wait time is larger than a constant. Second, we bound the robability that the adjusted wait time is larger than a constant. This is done in three stes, first the robability is bounded by two terms, then, each of these is bounded searately. We then comare the two robabilities for all i.

6 .4 N=5,, sigma=, delta=a Mean ercent adjustment between nominal and adjusted arrival times Number of Different Signals ) Figure 6: The average relative adjustment made when each customer listens to their signals comletely, N y i) x i N i= x i, as varies from to 5. As the number of signals er eriod,, increases the average adjustment also increases. We introduced eriodic signals to avoid large adjustments such that customers would only have to make reasonable adjustments to arrive at their signals. Observe that should be chosen small enough such that these adjustments are reasonable. Part I. The first art of the roof is devoted to P d i > ζ + z d i = z). First, observe that conditioned on the event d i = z, we have d i = max{a + x i + z x i, }. We assume that the nominal arrival times, x i, are determined by a nonhomogenous Poisson rocess, where the exected number of arrivals er unit time varies with time. The nominal arrival times are indeendent of each other and will be Poisson distributed. We define e i as the difference between two consecutive nominal arrival times; x i = x i + e i. We assume that x = such that x i = i j= ej. The ei are indeendent of each other and exonentially distributed with the arameter r. We rewrite our wait time using e i. d i = max{a + x i + z x i, } = d i = max{a + z e i, } We aim to find the robability that the FIFO wait time will be greater than a constant. Since ζ + z >, we can write P d i > ζ + z d i = z) = P a + z e i > ζ + z a + z e i > ) = P e i < a ζ e i < a + z) = P e i < a ζ) = e λa ζ). Part II. The second art of the roof is devoted to P h i > ζ + z h i = z). Figure 7: Burst atterns with different numbers R of customers er burst, for R = N/4, N/8, N/, N/4. When R is large, then there are a small number of arrival bursts but the eriod is very large. When R is small, there there are lots of quick bursts of arrivals. Observe that like d i, the new wait time h i deends on z, the variable y i), and the service time a: h i = max{a + y i ) + z y i), }. Since ζ + z >, we can write P h i > ζ + z h i = z) 3) = P max{a + y i ) + z y i), } > ζ + z ) 4) = P y i ) y i) > ζ a ). 5) Recall that the non-ordered adjusted arrival time is y i = σ i)x i + σ i xi + si ) There exists an index that deends on the index of the ordered arrival times, ki), such that y i) = σ ki) )x ki) + σ ki) x ki) + s ki) ) There exists another index ki ) that deends on the index of the ordered arrival times, such that y i ) = σ ki ) )x ki ) For ease of notation, we write: + σ ki ) x ki ) + s ki ) ) α i = σ ki ) )x ki ) σ ki) )x ki) β i = σ ki ) x ki ) y i ) y i) = α i + β i. We can rewrite 5) as: + s ki ) P h i > ζ + z h i = z) = P α i + β i > ζ a). ) σ ki) x ki) + s ki) ) By introducing an arbitrary constant γ and conditioning, we obtain: P α i + β i > ζ a) = P β i > ζ a α i α i < γ}p{α i < γ) + P β i > ζ a α i α i γ) P α i γ).

7 N=5,, sigma=, delta=a N=5,, sigma=, delta=a Relative Performance Rush Period = N/4 Rush Period = N/ Rush Period = N/8 Rush Period = N/4 5 5 Number of Different Signals ) Figure 8: Comaring the relative erformances of different burst atterns as varies from to,. The number of customers arriving during a burst is varied; R = N, N, N, N. When R is small, then the burst is also small. should be big enough such that the signal eriod is bigger than the burst. By observing that P β i > ζ a α i α i γ) Relative Performance Rush Rate >> Normal Rate Rush Rate > Normal Rate Rush Rate = Normal Rate 5 5 Number of Different Signals ) Figure 9: Comaring the relative erformances of different arrival rates as varies from to,. We defined two values for the arrival rate; a large λ, and a small λ. When λ is much larger than λ, our system erforms very well and decreases the wait times by a lot. When λ is slightly larger than λ, the system erforms well. In the extreme case, when ˆλ = λ, our system does not decrease wait times by very much. and that P β i > ζ a α i α i < γ) P β i > ζ a γ), we obtain: P h i > ζ + z h i = z) 6) P β i > ζ a γ) + P α i γ). 7) Part III. Now, we bound the second term on the right-hand side of 7): P α i γ) 8) P σ ki ) )x ki ) γ ) 9) P x ki ) γ ) ) P x N γ) ) P x N N/λ γ N/λ) ) P x N N/λ γ N/λ) λ ) N 3) N λ Nλ γ N/λ), 4) where we used the facts that α i is the difference of two nonnegative valued random variable, and that σ ki), and the last inequality follows by Chebychev s Inequality. Part IV. Now, we turn our attention to: P β i > ζ a γ) = Pσ ki ) x ki ) + s ki ) ) σ ki) x ki) P σ ki ) x ki ) P + P + s ki) ) > ζ a γ) + s ki ) ) > ζ a γ ) σ ki ) x ki ) > ζ a γ s ki ) σ ki ) > ζ a γ ). where we used the Union Bound to obtain the last inequality. Next, we consider the two terms on the right-hand side searately. First, observe that, by similar arguments as Part III, P σ ki ) x ki ) > ζ a γ ) P x ki ) > ζ a γ ) ) 5) 6) Nλ ζ a γ)/ N/λ). 7)

8 Relative Performance N=5,, delta=a Sigma= Sigma= Beta Dist. a=b=5 Beta Dist. a=5, b= Beta Dist. a=5, b= Probability Density Function of Beta Distributions a=b=5 a=5, b= a=5, b= Number of Different Signals ) Figure : Comaring the relative erformances of different distributions of the number of customers listen to their signals, σ, as varies from to,. The ideal case is when all of the customers listen to their signals, σ =. The worst case is when nobody listens to their signals, σ =. Realistically, not every customer is going to follow their signals exactly. We choose σ to be a beta distribution. We examine the erformances of three different beta distributions and all erform better than when σ =. The lots of the beta distributions robability density functions are shown in Figure. The system erforms best when the mean of the beta distribution is close to one. Finally, we bound: s ki ) P σ ki ) s ki ) = P σ ki ) > ζ a γ ) µµ > ζ a γ µ µ ) 8) 9) v ζ a γ µ, ) µ ) where we used the indeendence roerty and Chebychev s Inequality, and the definitions of v, µ, and µ. The claim of ) follows by combining 4), 7), and ). B. REFERENCES [] R. Adams. Active queue management: a survey. Communications Surveys & Tutorials, IEEE, 53):45 476, 3. [] C. E. Agnew. Dynamic modeling and control of congestion-rone systems. Oerations research, 43):4 49, 976. [3] E. Altman. Alication of dynamic games in queues. Annals of the International Society of Dynamic Games, ages 39 34, 5. [4] F. Antunes, F. Coito, and H. Duarte-Ramos. A linear aroach towards modeling human behavior. Technological Innovation for Sustainability, ages 35 34,. Figure : Probability density functions of different beta distributions that are used as the distribution of σ in Figure. [5] M. Armony, N. Shimkin, and W. Whitt. The imact of delay announcements in many-server queues with abandonment. Oerations Research, ages 66 8, 9. [6] J. Blanchet, P. Glynn, and S. Meyn. Large deviations for the emirical mean of an M/M/ queue. To aear in Queueing Systems, Secial Issue on Simulation of Networks,. [7] M. Chiang, A. Sutivong, and S. Boyd. Efficient nonlinear otimizations of queuing systems. In Global Telecommunications Conference, volume 3, ages 45 49,. [8] W. G. Cochran. Samling Techniques. Wiley, 977. [9] R. J. Gibbens and F. P. elly. On acket marking at riority queues. IEEE Transactions on Automatic Control, 47:6,. [] P. W. Glynn and W. Whitt. A central-limit-theorem version of l = λw. Queueing Systems, :9 5, 986. [] M. Harchol-Balter. Performance modeling and design of comuter systems: queueing theory in action. Cambridge University Press, 3. [] M. Harchol-Balter, T. Osogami, A. Scheller-Wolf, and A. Wierman. Multi-server queueing systems with multile riority classes. Queueing Systems, 53-4):33 36, 5. [3] R. Hassin and M. Haviv. To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems. Sringer, 3. [4] S. olli and. arlaalem. Mama: Multi-agent management of crowds to avoid stamedes in long queues. In Proceedings of AAMAS, ages 3 4, 3. [5] P. Naor. The regulation of queue size by levying tolls. Econometrica, 37:5 4, 969. [6] M. J. Neely. Stochastic network otimization with alication to communication and queueing systems. Synthesis Lectures on Communication Networks, 3):,.

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