We study reflecting Brownian motion with drift constrained to a wedge in the plane. Our first set... more We study reflecting Brownian motion with drift constrained to a wedge in the plane. Our first set of results provide necessary and sufficient conditions for existence and uniqueness of a solution to the corresponding submartingale problem with drift, and show that its solution possesses the Markov and Feller properties. Next, we study a version of the problem with absorption at the vertex of the wedge. In this case, we provide a condition for existence and uniqueness of a solution to the problem and some results on the probability of the vertex being reached.
The scores were statistically worse up to 3 months (p!.0001) and at 3 months (p 5 .001) compared ... more The scores were statistically worse up to 3 months (p!.0001) and at 3 months (p 5 .001) compared to baseline, but there was no difference between scores at baseline vs. 6 months (p 5 .60). Clinically, patients had returned to baseline by 3 months (difference in EPIC score of !10 points). This exact same pattern was noted when comparing scores within the subscales of urinary function, bother, incontinence, and irritative/obstructive symptoms. No factor, including prostate volume, D90, V100, V150, V200, total number of seeds, total number of needles, urethral length, dose to bladder neck, or worse pre-implant urinary function predicted for urinary toxicity at 6 months. Only a worse pre-implant urinary function predicted for increased toxicity at 3 months (p 5 .001). Conclusions: Patients undergoing PB with 131 Cs experience the most severe urinary toxicity 2-4 weeks following the implant with recovery from urinary toxicity to clinical baseline by 3 months and statistical baseline by 6 months. This appears to be quicker than historical controls who undergo PB with 125 I. On multivariate analysis, the only factor that predicts for urinary toxicity at 3 months is poor pre-implant function. This is consistent with historical results.
Motivated by practices in modern supply chains, we consider capacity-inventory joint management f... more Motivated by practices in modern supply chains, we consider capacity-inventory joint management for a make-to-stock manufacturing system operating under a base stock policy. The production facility is modeled as multiple servers operating in parallel. The number of servers corresponds to the capacity decision and the base stock level is the inventory decision. The main problem which we consider is the joint optimization of the capacity and inventory decisions to minimize a combination of capacity, inventory, and backordering costs. We develop a square-root rule for the joint decision and justify the rule analytically in a many-server queue asymptotic framework. We also provide operational insights into the tradeoffs involved in such joint management problems, through various analysis based on the square-root rule as well as a comparison with analogous results for single-server make-to-stock queues.
We study a single-server queue, operating under the first-in-first-out (FIFO) service discipline,... more We study a single-server queue, operating under the first-in-first-out (FIFO) service discipline, in which each customer independently abandons the queue if his service has not begun within a generally distributed amount of time. Under some mild conditions on the abandonment distribution, we identify a limiting heavy-traffic regime in which the resulting diffusion approximation for both the offered waiting time process (the process that tracks the amount of time an infinitely patient arriving customer would wait for service) and the queue-length process contain the entire abandonment distribution. To use a continuous mapping approach to establish our weak convergence results, we additionally develop existence, uniqueness, and continuity results for nonlinear generalized regulator mappings that are of independent interest. We further perform a simulation study to evaluate the quality of the proposed approximations for the steady-state mean queue length and the steady-state probability of abandonment suggested by the limiting diffusion process.
Bid and ask sizes at the top of the order book provide information on short-term price moves. Dra... more Bid and ask sizes at the top of the order book provide information on short-term price moves. Drawing from classical descriptions of the order book in terms of queues and orderarrival rates (Smith et al (2003)), we consider a diffusion model for the evolution of the best bid/ask queues. We compute the probability that the next price move is upward, conditional on the best bid/ask sizes, the hidden liquidity of the market and the correlation between changes in the bid/ask sizes. The model can be useful, among other things, to rank trading venues in terms of the "information content" of their quotes and to estimate the hidden liquidity in a market based on high-frequency data. We illustrate the approach with an empirical study of a few liquid stocks using quotes from various exchanges.
background Avelumab, a human anti-programmed death-ligand 1 immunoglobulin G1 monoclonal antibody... more background Avelumab, a human anti-programmed death-ligand 1 immunoglobulin G1 monoclonal antibody, showed favorable efficacy and safety in patients with metastatic Merkel cell carcinoma (mMCC) in the phase II JAVELIN Merkel 200 trial, leading to approval in multiple countries. We describe real-world experience with avelumab in patients with mMCC from an expanded access program. Methods Eligible patients had mMCC and progressive disease during or after chemotherapy or were ineligible for chemotherapy or clinical trial participation. Patients received an initial 3-month supply of avelumab (administered as 10 mg/kg intravenously every 2 weeks until progressive disease or unacceptable toxicity); resupply was allowed following complete response, partial response, stable disease, or clinical benefit per physician assessment. results Between December 15, 2015, and March 4, 2019, 558 of 620 requests from 38 countries were medically approved, and 494 patients received avelumab. Among 240 evaluable patients, the objective response rate was 46.7% (complete response in 22.9%, including 3 of 16 potentially immunocompromised patients), and the disease control rate was 71.2%. The median duration of treatment in evaluable patients with response was 7.9 months (range, 1.0-41.7) overall and 5.2 months (range, 3.0-13.9) in immunocompromised patients. No new safety signals were identified. The expanded access program closed for new requests on December 31, 2018, as required after regulatory approval; benefitting patients continued to receive avelumab. Conclusions The avelumab expanded access program for patients with mMCC demonstrated efficacy and safety in a real-world setting, consistent with the results from JAVELIN Merkel 200, and provided a treatment for patients with limited options.
We study the G/GI/∞ queue in heavy-traffic using tempered distribution-valued processes which tra... more We study the G/GI/∞ queue in heavy-traffic using tempered distribution-valued processes which track the age and residual service time of each customer in the system. In both cases, we use the continuous mapping theorem together with functional central limit theorem results in order to obtain fluid and diffusion limits for these processes in the space of tempered distributionvalued processes. We find that our diffusion limits are tempered distribution-valued Ornstein-Uhlenbeck processes.
In this paper, we study a reflected AR(1) process, i.e., a process (Z n) n obeying the recursion ... more In this paper, we study a reflected AR(1) process, i.e., a process (Z n) n obeying the recursion Z n+1 = max{aZ n + X n , 0}, with (X n) n a sequence of i.i.d. random variables. We find explicit results for the distribution of Z n (in terms of transforms) in case X n can be written as Y n − B n , with (B n) n being a sequence of independent random variables which are all exp(λ) distributed, and (Y n) n i.i.d.; when |a| < 1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (B n) n are of phasetype. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein-Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a Normal random variable conditioned on being positive. KEYWORDS. Reflected processes queueing scaling limits
In this paper, we consider a single-server queue fed by K independent renewal arrival streams, ea... more In this paper, we consider a single-server queue fed by K independent renewal arrival streams, each representing a different job class. Jobs are processed in a FIFO fashion, regardless of class. The total amount of work arriving to the system exceeds the server's capacity. That is, the nominal traffic intensity of the system is assumed to be greater than one. Jobs arriving to the system grow impatient and abandon the queue after a random amount of time if service has not yet begun. Interarrival, service and abandonment times are assumed to be generally distributed and class specific. We approximate this system using both fluid and diffusion limits. To this end, we consider a sequence of systems indexed by n in which the arrival and service rates are proportional to n; the abandonment distribution remains fixed across the sequence. In our first main result, we show that in the limit as n tends to infinity, the virtual waiting time process for those jobs that do not abandon from the queue converges to a limiting deterministic process. This limit may be characterized as the solution to a first order ODE. Specific examples are then presented for which the ODE may be explicitly solved. In our second main result, we refine the deterministic fluid approximation by showing that the fluid centered and diffusion scaled virtual waiting time process weakly converges to an Ornstein-Uhlenbeck process whose drift and infinitesimal variance both vary over time. This process may also be solved for explicitly, thus yielding approximations to the transient as well as steady state behavior of the queue.
We establish a heavy-traffic limit theorem on convergence in distribution for the number of custo... more We establish a heavy-traffic limit theorem on convergence in distribution for the number of customers in a many-server queue when the number of servers tends to infinity. No critical loading condition is assumed. Generally, the limit process does not have trajectories in the Skorohod space. We give conditions for the convergence to hold in the topology of compact convergence. Some new results for an infinite server are also provided.
Bandwidth-sharing networks as introduced by Massoulié & Roberts (1998) model the dynamic interact... more Bandwidth-sharing networks as introduced by Massoulié & Roberts (1998) model the dynamic interaction among an evolving population of elastic flows competing for several links. With policies based on optimization procedures, such models are of interest both from a Queueing Theory and Operations Research perspective. In the present paper, we focus on bandwidth-sharing networks with capacities and arrival rates of a large order of magnitude compared to transfer rates of individual flows. This regime is standard in practice. In particular, we extend previous work by Reed & Zwart (2010) on fluid approximations for such networks: we allow interarrival times, flow sizes and patient times (i.e. abandonment times measured from the arrival epochs) to be generally distributed, rather than exponentially distributed. We also develop polynomial-time computable fixed-point approximations for stationary distributions of bandwidth-sharing networks, and suggest new techniques for deriving these types of results.
We consider a stochastic differential equation (SDE) with piecewise linear drift driven by a spec... more We consider a stochastic differential equation (SDE) with piecewise linear drift driven by a spectrally one-sided Lévy process. We show that this SDE has some connections with queueing and storage models, and we use this observation to obtain the invariant distribution.
We study queues in tandem with customer deadlines and retrials. We first consider a 2-queue Marko... more We study queues in tandem with customer deadlines and retrials. We first consider a 2-queue Markovian system with blocking at the second queue, analyze it, and derive its stability condition. We then study a non-Markovian setting and derive the stability condition for an approximating diffusion, showing its similarity to the former condition. In the Markovian setting, we use probability generating functions and matrix analytic techniques. In the diffusion setting, we consider expectations of the first hitting times of compact sets.
Bid and ask sizes at the top of the order book provide information on short-term price moves. Dra... more Bid and ask sizes at the top of the order book provide information on short-term price moves. Drawing from classical descriptions of the order book in terms of queues and order-arrival rates (Smith et al., 2003), we consider a diffusion model for the evolution of the best bid/ask queues. We compute the probability that the next price move is upward, conditional on the best bid/ask sizes, the hidden liquidity in the market and the correlation between changes in the bid/ask sizes. The model can be useful, among other things, to rank trading venues in terms of the "information content" of their quotes and to estimate hidden liquidity in a market based on high-frequency data. We illustrate the approach with an empirical study of a few stocks using quotes from various exchanges.
We study random walks whose increments are α-stable distributions with shape parameter 1 < α < 2.... more We study random walks whose increments are α-stable distributions with shape parameter 1 < α < 2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to zero and, in the totally skewed to the left case of skewness parameter β = −1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables and Zolotarev's integral representation for the CDF of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.
Can you sell multiple items by providing only prices for different sizes of bundles rather than t... more Can you sell multiple items by providing only prices for different sizes of bundles rather than the different possible combinations of them? In this paper, we provide a framework for understanding “bundle-size pricing” (or simply, BSP) where only a menu of bundle sizes and their corresponding prices are offered. Although BSP is commonly used across several industries, little is known about the optimal BSP policy in terms of sizes and prices, along with the theoretical properties of its profit. In this paper, we provide a simple and tractable theoretical framework to analyze the large-scale BSP problem where a multiproduct firm is selling a large number of products. We characterize the circumstances under which such policies perform well by studying the effect of various factors such as marginal cost or customers’ budget on the performance of BSP and identify possible causes of its inefficiency.
In this paper, we study the G/GI/N queue in the Halfin-Whitt regime. Our first result is to obtai... more In this paper, we study the G/GI/N queue in the Halfin-Whitt regime. Our first result is to obtain a deterministic fluid limit for the properly centered and scaled number of customers in the system which may be used to provide a first-order approximation to the queue length process. Our second result is to obtain a second-order stochastic approximation to the number of customers in the system in the Halfin-Whitt regime. This is accomplished by first centering the queue length process by its deterministic fluid limit and then normalizing by an appropriate factor. We then proceed to obtain an alternative but equivalent characterization of our limiting approximation which involves the renewal function associated with the service time distribution. This alternative characterization reduces to the diffusion process obtained by Halfin and Whitt [Oper. Res. 29 (1981) 567-588] in the case of exponentially distributed service times.
We study reflecting Brownian motion with drift constrained to a wedge in the plane. Our first set... more We study reflecting Brownian motion with drift constrained to a wedge in the plane. Our first set of results provide necessary and sufficient conditions for existence and uniqueness of a solution to the corresponding submartingale problem with drift, and show that its solution possesses the Markov and Feller properties. Next, we study a version of the problem with absorption at the vertex of the wedge. In this case, we provide a condition for existence and uniqueness of a solution to the problem and some results on the probability of the vertex being reached.
The scores were statistically worse up to 3 months (p!.0001) and at 3 months (p 5 .001) compared ... more The scores were statistically worse up to 3 months (p!.0001) and at 3 months (p 5 .001) compared to baseline, but there was no difference between scores at baseline vs. 6 months (p 5 .60). Clinically, patients had returned to baseline by 3 months (difference in EPIC score of !10 points). This exact same pattern was noted when comparing scores within the subscales of urinary function, bother, incontinence, and irritative/obstructive symptoms. No factor, including prostate volume, D90, V100, V150, V200, total number of seeds, total number of needles, urethral length, dose to bladder neck, or worse pre-implant urinary function predicted for urinary toxicity at 6 months. Only a worse pre-implant urinary function predicted for increased toxicity at 3 months (p 5 .001). Conclusions: Patients undergoing PB with 131 Cs experience the most severe urinary toxicity 2-4 weeks following the implant with recovery from urinary toxicity to clinical baseline by 3 months and statistical baseline by 6 months. This appears to be quicker than historical controls who undergo PB with 125 I. On multivariate analysis, the only factor that predicts for urinary toxicity at 3 months is poor pre-implant function. This is consistent with historical results.
Motivated by practices in modern supply chains, we consider capacity-inventory joint management f... more Motivated by practices in modern supply chains, we consider capacity-inventory joint management for a make-to-stock manufacturing system operating under a base stock policy. The production facility is modeled as multiple servers operating in parallel. The number of servers corresponds to the capacity decision and the base stock level is the inventory decision. The main problem which we consider is the joint optimization of the capacity and inventory decisions to minimize a combination of capacity, inventory, and backordering costs. We develop a square-root rule for the joint decision and justify the rule analytically in a many-server queue asymptotic framework. We also provide operational insights into the tradeoffs involved in such joint management problems, through various analysis based on the square-root rule as well as a comparison with analogous results for single-server make-to-stock queues.
We study a single-server queue, operating under the first-in-first-out (FIFO) service discipline,... more We study a single-server queue, operating under the first-in-first-out (FIFO) service discipline, in which each customer independently abandons the queue if his service has not begun within a generally distributed amount of time. Under some mild conditions on the abandonment distribution, we identify a limiting heavy-traffic regime in which the resulting diffusion approximation for both the offered waiting time process (the process that tracks the amount of time an infinitely patient arriving customer would wait for service) and the queue-length process contain the entire abandonment distribution. To use a continuous mapping approach to establish our weak convergence results, we additionally develop existence, uniqueness, and continuity results for nonlinear generalized regulator mappings that are of independent interest. We further perform a simulation study to evaluate the quality of the proposed approximations for the steady-state mean queue length and the steady-state probability of abandonment suggested by the limiting diffusion process.
Bid and ask sizes at the top of the order book provide information on short-term price moves. Dra... more Bid and ask sizes at the top of the order book provide information on short-term price moves. Drawing from classical descriptions of the order book in terms of queues and orderarrival rates (Smith et al (2003)), we consider a diffusion model for the evolution of the best bid/ask queues. We compute the probability that the next price move is upward, conditional on the best bid/ask sizes, the hidden liquidity of the market and the correlation between changes in the bid/ask sizes. The model can be useful, among other things, to rank trading venues in terms of the "information content" of their quotes and to estimate the hidden liquidity in a market based on high-frequency data. We illustrate the approach with an empirical study of a few liquid stocks using quotes from various exchanges.
background Avelumab, a human anti-programmed death-ligand 1 immunoglobulin G1 monoclonal antibody... more background Avelumab, a human anti-programmed death-ligand 1 immunoglobulin G1 monoclonal antibody, showed favorable efficacy and safety in patients with metastatic Merkel cell carcinoma (mMCC) in the phase II JAVELIN Merkel 200 trial, leading to approval in multiple countries. We describe real-world experience with avelumab in patients with mMCC from an expanded access program. Methods Eligible patients had mMCC and progressive disease during or after chemotherapy or were ineligible for chemotherapy or clinical trial participation. Patients received an initial 3-month supply of avelumab (administered as 10 mg/kg intravenously every 2 weeks until progressive disease or unacceptable toxicity); resupply was allowed following complete response, partial response, stable disease, or clinical benefit per physician assessment. results Between December 15, 2015, and March 4, 2019, 558 of 620 requests from 38 countries were medically approved, and 494 patients received avelumab. Among 240 evaluable patients, the objective response rate was 46.7% (complete response in 22.9%, including 3 of 16 potentially immunocompromised patients), and the disease control rate was 71.2%. The median duration of treatment in evaluable patients with response was 7.9 months (range, 1.0-41.7) overall and 5.2 months (range, 3.0-13.9) in immunocompromised patients. No new safety signals were identified. The expanded access program closed for new requests on December 31, 2018, as required after regulatory approval; benefitting patients continued to receive avelumab. Conclusions The avelumab expanded access program for patients with mMCC demonstrated efficacy and safety in a real-world setting, consistent with the results from JAVELIN Merkel 200, and provided a treatment for patients with limited options.
We study the G/GI/∞ queue in heavy-traffic using tempered distribution-valued processes which tra... more We study the G/GI/∞ queue in heavy-traffic using tempered distribution-valued processes which track the age and residual service time of each customer in the system. In both cases, we use the continuous mapping theorem together with functional central limit theorem results in order to obtain fluid and diffusion limits for these processes in the space of tempered distributionvalued processes. We find that our diffusion limits are tempered distribution-valued Ornstein-Uhlenbeck processes.
In this paper, we study a reflected AR(1) process, i.e., a process (Z n) n obeying the recursion ... more In this paper, we study a reflected AR(1) process, i.e., a process (Z n) n obeying the recursion Z n+1 = max{aZ n + X n , 0}, with (X n) n a sequence of i.i.d. random variables. We find explicit results for the distribution of Z n (in terms of transforms) in case X n can be written as Y n − B n , with (B n) n being a sequence of independent random variables which are all exp(λ) distributed, and (Y n) n i.i.d.; when |a| < 1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (B n) n are of phasetype. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein-Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a Normal random variable conditioned on being positive. KEYWORDS. Reflected processes queueing scaling limits
In this paper, we consider a single-server queue fed by K independent renewal arrival streams, ea... more In this paper, we consider a single-server queue fed by K independent renewal arrival streams, each representing a different job class. Jobs are processed in a FIFO fashion, regardless of class. The total amount of work arriving to the system exceeds the server's capacity. That is, the nominal traffic intensity of the system is assumed to be greater than one. Jobs arriving to the system grow impatient and abandon the queue after a random amount of time if service has not yet begun. Interarrival, service and abandonment times are assumed to be generally distributed and class specific. We approximate this system using both fluid and diffusion limits. To this end, we consider a sequence of systems indexed by n in which the arrival and service rates are proportional to n; the abandonment distribution remains fixed across the sequence. In our first main result, we show that in the limit as n tends to infinity, the virtual waiting time process for those jobs that do not abandon from the queue converges to a limiting deterministic process. This limit may be characterized as the solution to a first order ODE. Specific examples are then presented for which the ODE may be explicitly solved. In our second main result, we refine the deterministic fluid approximation by showing that the fluid centered and diffusion scaled virtual waiting time process weakly converges to an Ornstein-Uhlenbeck process whose drift and infinitesimal variance both vary over time. This process may also be solved for explicitly, thus yielding approximations to the transient as well as steady state behavior of the queue.
We establish a heavy-traffic limit theorem on convergence in distribution for the number of custo... more We establish a heavy-traffic limit theorem on convergence in distribution for the number of customers in a many-server queue when the number of servers tends to infinity. No critical loading condition is assumed. Generally, the limit process does not have trajectories in the Skorohod space. We give conditions for the convergence to hold in the topology of compact convergence. Some new results for an infinite server are also provided.
Bandwidth-sharing networks as introduced by Massoulié & Roberts (1998) model the dynamic interact... more Bandwidth-sharing networks as introduced by Massoulié & Roberts (1998) model the dynamic interaction among an evolving population of elastic flows competing for several links. With policies based on optimization procedures, such models are of interest both from a Queueing Theory and Operations Research perspective. In the present paper, we focus on bandwidth-sharing networks with capacities and arrival rates of a large order of magnitude compared to transfer rates of individual flows. This regime is standard in practice. In particular, we extend previous work by Reed & Zwart (2010) on fluid approximations for such networks: we allow interarrival times, flow sizes and patient times (i.e. abandonment times measured from the arrival epochs) to be generally distributed, rather than exponentially distributed. We also develop polynomial-time computable fixed-point approximations for stationary distributions of bandwidth-sharing networks, and suggest new techniques for deriving these types of results.
We consider a stochastic differential equation (SDE) with piecewise linear drift driven by a spec... more We consider a stochastic differential equation (SDE) with piecewise linear drift driven by a spectrally one-sided Lévy process. We show that this SDE has some connections with queueing and storage models, and we use this observation to obtain the invariant distribution.
We study queues in tandem with customer deadlines and retrials. We first consider a 2-queue Marko... more We study queues in tandem with customer deadlines and retrials. We first consider a 2-queue Markovian system with blocking at the second queue, analyze it, and derive its stability condition. We then study a non-Markovian setting and derive the stability condition for an approximating diffusion, showing its similarity to the former condition. In the Markovian setting, we use probability generating functions and matrix analytic techniques. In the diffusion setting, we consider expectations of the first hitting times of compact sets.
Bid and ask sizes at the top of the order book provide information on short-term price moves. Dra... more Bid and ask sizes at the top of the order book provide information on short-term price moves. Drawing from classical descriptions of the order book in terms of queues and order-arrival rates (Smith et al., 2003), we consider a diffusion model for the evolution of the best bid/ask queues. We compute the probability that the next price move is upward, conditional on the best bid/ask sizes, the hidden liquidity in the market and the correlation between changes in the bid/ask sizes. The model can be useful, among other things, to rank trading venues in terms of the "information content" of their quotes and to estimate hidden liquidity in a market based on high-frequency data. We illustrate the approach with an empirical study of a few stocks using quotes from various exchanges.
We study random walks whose increments are α-stable distributions with shape parameter 1 < α < 2.... more We study random walks whose increments are α-stable distributions with shape parameter 1 < α < 2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to zero and, in the totally skewed to the left case of skewness parameter β = −1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables and Zolotarev's integral representation for the CDF of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.
Can you sell multiple items by providing only prices for different sizes of bundles rather than t... more Can you sell multiple items by providing only prices for different sizes of bundles rather than the different possible combinations of them? In this paper, we provide a framework for understanding “bundle-size pricing” (or simply, BSP) where only a menu of bundle sizes and their corresponding prices are offered. Although BSP is commonly used across several industries, little is known about the optimal BSP policy in terms of sizes and prices, along with the theoretical properties of its profit. In this paper, we provide a simple and tractable theoretical framework to analyze the large-scale BSP problem where a multiproduct firm is selling a large number of products. We characterize the circumstances under which such policies perform well by studying the effect of various factors such as marginal cost or customers’ budget on the performance of BSP and identify possible causes of its inefficiency.
In this paper, we study the G/GI/N queue in the Halfin-Whitt regime. Our first result is to obtai... more In this paper, we study the G/GI/N queue in the Halfin-Whitt regime. Our first result is to obtain a deterministic fluid limit for the properly centered and scaled number of customers in the system which may be used to provide a first-order approximation to the queue length process. Our second result is to obtain a second-order stochastic approximation to the number of customers in the system in the Halfin-Whitt regime. This is accomplished by first centering the queue length process by its deterministic fluid limit and then normalizing by an appropriate factor. We then proceed to obtain an alternative but equivalent characterization of our limiting approximation which involves the renewal function associated with the service time distribution. This alternative characterization reduces to the diffusion process obtained by Halfin and Whitt [Oper. Res. 29 (1981) 567-588] in the case of exponentially distributed service times.
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Papers by Josh Reed