Yuedong Xu
email: ydxu@fudan.edu.cn

Fellow, IEEE Eitan Altman
email: eitan.altman@inria.fr

Rachid El-Azouzi
email: rachid.elazouzi@univ-avignon.fr

Majed Haddad

Salaheddine Elayoubi
email: salahed-dine.elayoubi@orange.com

Tania Jimenez
email: tania.jimenez@univ-avignon.fr

Analysis of Buffer Starvation with Application to Objective QoE Optimization of Streaming Services

Keywords: Starvation, Start-up Delay, Quality of Experience, Ballot Theorem EDICS: 8-MSAT, 3-QAUE

Our purpose in this paper is to characterize buffer starvations for streaming services. The buffer is modeled as a FIFO queue with exponential service time and Poisson arrivals. When the buffer is empty, the service restarts after a certain amount of packets are prefetched. With this goal, we propose two approaches to obtain the exact distribution of the number of buffer starvations, one of which is based on Ballot theorem, and the other uses recursive equations. The Ballot theorem approach gives an explicit result. We extend this approach to the scenario with a constant playback rate using Tàkacs Ballot theorem. The recursive approach, though not offering an explicit result, allows us to obtain the distribution of starvations with non-independent and identically distributed (i.i.d.) arrival process in which an ON/OFF bursty arrival process is considered in this work.We further compute the starvation probability as a function of the amount of prefetched packets for a large number of files via a fluid analysis. Among many potential applications of starvation analysis, we show how to apply it to optimize the objective quality of experience (QoE) of media streaming, by exploiting the tradeoff between startup/rebuffering delay and starvations.

I. INTRODUCTION

The starvation probability of a buffer is an important performance measure for protocol design of telecommunication networks, as well as in storage systems and ecological systems (e.g. dams). Starvation is said to occur when the buffer is empty. Various applications use buffering in order to control the rate at which packets are served at the destination. As long as there are packets in the buffer, packets arrive at the destination regularly, i.e. they are spaced by the service time of the buffer. Once the buffer empties packets may arrive at the destination separated by larger times, as the spacing between packets now depends also on the inter-arrival times at the queue. Starvation is in particular undesirable in video streaming applications.

The time till starvation of a queue is related to the busy period which has been well studied under the assumption of a stationary arrival process (see [START_REF] Baccelli | A Sample Path Analysis of the M/M/1 Queue[END_REF], [START_REF] Ledermann | Spectral Theory for the Differential Equations of Simple Birth and Death Processes[END_REF] and their references). In contrast to this assumption, we consider a finite number of arrivals as we are interested in statistics of starvation when a file of fixed size is transferred.

The main goal of this paper is to find the distribution of the number of starvations within a file of N packets. We first model the buffer as an M/M/1 queue, and then extend it to incorporate the bursty packet arrival that is modeled by an interrupted Poisson process (IPP). In this system, a fixed amount of packets are prefetched (also called "prefetching threshold") before the service begins or resumes after a starvation event.

In this paper, we mainly propose two approaches (that give the same result) to compute the distribution of the number of starvations for a single file. The first approach gives an explicit result based on the famous Ballot theorem [START_REF] Takacs | Ballot problems[END_REF]. But it is in general suitable for independent and identically distributed (i.i.d.) arrival process. This motivates us to propose the second approach that yields a more flexible recursive algorithm. Although it has no explicit result, it can be used to compute the starvation probabilities for more complicated arrival processes. In terms of complexity, the recursive approach can compute the starvation probabilities for different combinations of the initial start-up threshold and the file size in one round, which means an overall smaller complexity than the Ballot theorem approach.

The key feature of Ballot theorem is its simple expression to compute the probability that a counting process (e.g. arrival process) is strictly ahead of another counting process (e.g. departure process). Using Ballot theorem, we can compute in a simple way the exact distribution of the number of starvations explicitly. When the file size is large enough, we present the asymptotic starvation probability using Gaussian (interchangeable with Normal) approximation as well as an approximation of the Riemann November 30, 2013 DRAFT integral. As a special case that the playout buffer is modeled by an M/D/1 queue, the probabilities of the number of starvations can be obtained on the basis of a discrete version of Ballot theorem. Furthermore, unlike the Ballot theorem that usually requires the i.i.d. packet arrivals, the recursive approach enables us to compute the probability of starvations with continuous-time ON/OFF bursty packet arrivals. The media server transfers packets during the ON state and pauses during the OFF state. It switches from one state to the other after residing an exponentially distributed time period.

We further propose a fluid analysis of starvation behavior on the file level. This approach, instead of looking into the stochastic packet arrivals and departures, predicts the starvation where the servers manage a large quantity of file transfers. Given the traffic intensity and the distribution of file size, we are able to compute the starvation probability as a function of the prefetching threshold. The fluid analysis, though simple, offers an important insight on how to control the probability of starvation for many files, instead of for one particular file.

The probabilities of starvations developed in this work have various applications in the different fields. A prominent example is the media streaming service. This application demonstrates a dilemma between the prefetching process and the starvation. A longer prefetching process causes a larger startup/rebuffering delay, while a shorter one might result in starvations. The user perceived media quality (or QoE equivalently) is impaired by two major factors, the large start-up/rebuffering delay and the frequent starvations, according to the measurement studies in [START_REF] Dobrian | Understanding the Impact of Video Quality on User Engagement[END_REF], [START_REF] Krishnan | Video Stream Quality Impacts Viewer Behavior: Inferring Causality using Quasi-Experimental Designs[END_REF]. These two factors are thus defined as quality metrics of streaming service. The QoE study becomes increasingly important in the epoch that web video hogs up to more than 37% of total traffic during peak hours in USA [18]. In contrast to the rapid growth of traffic load, the bandwidth provision usually lags behind. In this context, media providers and network operators face a crucial challenge of maintaining a satisfactory QoE of streaming service.

With the results developed in this work, we are able to answer the fundamental question: How many packets should the media player prefetch to optimize the users' quality of experience?

To answer this question, we first use objective QoE costs to model the subjective human unhappiness for both finite and infinite file size. An objective QoE cost is the weighted sum functions on the start-up/rebuffering delay and the starvation behaviors. The weight reflects an individual user's relative impatience on the delay than on the starvations. We then formulate the objective QoE optimization problems in a variety of scenarios. After solving them, we obtain the optimal algorithms to configure the start-up/rebuffering thresholds in packets. Lastly, we discuss the possible implementation of the proposed algorithms. A streaming user can dynamically configure the rebuffering threshold. When a starvation happens, the packet arrival rate can be measured at the user side. Given the knowledge the arrival November 30, 2013 DRAFT rate, service rate and the remaining file size, the user can set the rebuffering threshold to balance the tradeoffs: i) delay vs starvation probability (section VI-A) for a small remaining file size and ii) delay vs starvation interval (section VI-B) for a large remaining file size. The media server can configure the start-up threshold for each category of videos (e.g. music, sports, and TV news) beforehand. Given the distributions of file size and the user throughput, it can use the algorithm in section VI-C to compute the optimal start-up threshold. Simulation studies in section VI demonstrate the relationships between the objective QoE costs and the optimal prefetching thresholds in a variety of scenarios.

We summarize the main results as follows.

• We propose a Ballot theorem approach to compute the distribution of the number of starvations for a file of finite size in an M/M/1 queue. It provides an explicit solution and is generalized to an M/D/1 queue.

• We propose a recursive approach to compute the distribution of the number of starvations. We further extend it to the ON/OFF bursty arrival process.

• We present a fluid model to compute the starvation probability, given the start-up threshold and the file size distributions of representative content types.

• Our study provides the important understandings on how the starvation probabilities are impacted by the prefetching threshold. We propose a set of strategies to configure better prefetching thresholds so as to balance the tradeoff between the delay and the starvations for video streaming services.

The rest of this paper is organized as follows. Section II reviews the related work. We propose a Ballot approach in Section III. Section IV presents the recursive approach for an M/M/1 queue and extents it to the ON/OFF bursty arrival process. Section V performs a fluid analysis for a large number of files.

Section VI presents the QoE metrics and their optimization issues. Our theoretical results are verified in section VII. Section VIII concludes this paper.

II. RELATED WORK

The analysis of starvation is close to that of busy period in transient queues. In [START_REF] Baccelli | A Sample Path Analysis of the M/M/1 Queue[END_REF], [START_REF] Ledermann | Spectral Theory for the Differential Equations of Simple Birth and Death Processes[END_REF] authors solve the distribution of the buffer size as a function of time for the M/M/1 queue. Ballot theorem and recursive equations have been used to analyze the packet loss probability in a finite buffer when the forward error-correcting technique is deployed. Citon et al. [START_REF] Citon | Analysis of packet loss processes in high-speed networks[END_REF] propose a recursive approach that enables them to compute the packet loss probability in a block of consecutive packet arrivals into an M/M/1/K queue. Based on their recursive approach, Altman and Jean-Marie in [START_REF] Altman | Loss probabilities for messages with redundant packets feeding a finite buffer[END_REF] obtain the expressions for the multidimensional generating function of the packet loss probability. The distribution of message delay November 30, 2013 DRAFT is given in an extended work [START_REF] Altman | The distribution of delays of dispersed messages in an M/M/1 queue[END_REF]. Dubea and Altman in [START_REF] Dubea | On loss probabilities in presence of redundant packets with random drop[END_REF] analyze the packet loss probability with the consideration of random loss in incoming and outgoing links. In [START_REF] Gurewitz | The Ballot Theorem Strikes Again: Packet Loss Process Distribution[END_REF], Gurewitz et al. introduce the powerful Ballot theorem to find this probability within a block of packet arrivals into an M/M/1/K queue.

They consider two cases, in which the block size is smaller or greater than the buffer limit. Another example of applying Ballot theorem to evaluate networking system is found in [START_REF] Humblet | Ballot theorems applied to the transient analysis of nD/D/1 queues[END_REF]. Humblet et al.

present a method based on Ballot theorem to study the performance of nD/D/1 queue with periodical arrivals and deterministic service time. In [START_REF] He | New Analysis Framework for Discrete Time Queueing Systems with General Stochastic Sources[END_REF], He and Sohraby use Ballot theorem to find the stationary probability distribution in a general class of discrete time systems. Privalov and Sohraby [START_REF] Privalova | Playout in Slotted CBR Networks: Single and Multiple Nodes[END_REF] study the underflow behavior of CBR traffic in a time-slotted queueing system. However, they do not provide the insights of having a certain number of starvations.

In the related applications related to our work, Stockhammer et al. [START_REF] Stockhammer | Streaming video over variable bit-rate wireless channels[END_REF] specify the minimum start-up delay and the minimum buffer size for a given video stream and a deterministic variable bit rate (VBR) wireless channel. Recently, [START_REF] Liang | Effect of delay and buffering on jitter-free streaming over random VBR channels[END_REF] presents a deterministic bound, and [START_REF] Parandehgheibi | Avoiding Interruptions a QoE Reliability Function for Streaming Media Applications[END_REF] provides a stochastic bound of start-up delay to avoid starvation. Authors in [START_REF] Hao Luan | Impact of network dynamics on users' video quality: analytical framework and QoS provision[END_REF] model the playout buffer as a G/G/1 queue. By using diffusion approximation, they obtain the closed-form starvation probability with asymptotically large file size. Xu et.al [START_REF] Xu | Cross-Layer Qos Scheduling for Layered Multicast Streaming in OFDMA Wireless Networks[END_REF] study the scheduling algorithms for multicast streaming in multicarrier wireless downlink. Authors in [START_REF] Xu | QoE Analysis of Media Streaming in Wireless Data Networks[END_REF] studied the QoE metrics of a persistent video streaming in cellular networks.

They further presented a new method in [START_REF] Xu | Impact of flow-level dynamics on QoE of video streaming in wireless networks[END_REF] to compute the QoE metrics for a cellular network with arrivals and departures of streaming flows. Our paper differs from some of the above works in the following ways: i) we present new theories that yield an exact probability of starvation, and the p.g.f. of starvation events; ii) we study the asymptotic behavior with error analysis; iii) we perform a macroscopic starvation analysis using a fluid model; iv) we configure optimal prefetching thresholds to optimize the QoE metrics.

III. STARVATION ANALYSIS USING BALLOT THEOREM

In this section, we study the starvation behavior of an M/M/1 (also extended to M/D/1) queue with finite number of arrivals based on the powerful Ballot theorem.

A. System Description

We consider a single media file with finite size N . The media content is pre-stored in the media server (e.g. video on demand (VoD) service). When a user makes a request, the server segments this media into packets, and transfers them to the user by use of TCP or UDP. When packets traverse the wired or November 30, 2013 DRAFT wireless links, their arrivals to the media player of a user are not deterministic due to the dynamics of the available bandwidth. The Poisson assumption is not the most realistic way to describe packet arrivals, but it reveals the essential features of the system, and is the first step for more general arrival processes.

After the streaming packets are received, they are first stored in the playout buffer. The interval between two packets that are served is assumed to be exponentially distributed so that we can model the receiver buffer as an M/M/1 queue. The maximum buffer size is assumed to be large enough so that the whole file can be stored. This simplification is justified by the fact that the storage space is usually very large in the receiver side (e.g. several GB).

The user perceived media quality has two measures called start-up delay and starvation. As explained earlier, the media player wants to avoid the starvation by prefetching packets. However, this action might incur a long waiting time. In what follows, we reveal the relationship between the start-up delay and the starvation behavior, with the consideration of file size.

B. A Packet Level Model

We present a packet level model to investigate the starvation behavior. We denote by λ the Poisson arrival rate of the packets, and by µ the Poisson service rate. We define ρ := λ/µ as the traffic intensity.

In a non-empty M/M/1 queue with everlasting arrivals, the rate at which either an arrival or a departure occurs is given by λ + µ. This event corresponds to an arrival with probability p, or is otherwise to an end of service with probability q, where

p = λ λ + µ = ρ 1 + ρ ; q = µ λ + µ = 1 1 + ρ .
The buffer is initially empty. We let T 1 be the start-up delay, in which x 1 packets are accumulated in the buffer. Our analysis of the probability of starvation is built on the famous Ballot theorem:

Ballot Theorem: In a ballot, candidate A scores N A votes and candidate B scores N B votes, where

N A > N B .
Assume that while counting, all the ordering (i.e. all sequences of A's and B's) are equally alike, the probability that throughout the counting, A is always ahead in the count of votes is NA-NB NA+NB . After the service begins, the probability of starvation is given by Theorem 1.

Theorem 1: For the initial queue length x 1 and the total size N of a file, the probability of starvation is given by:

P s = N -1 ∑ k=x1 x 1 2k -x 1 ( 2k -x 1 k -x 1 ) p k-x1 (1 -p) k . (1)
Proof: We define E k to be an event that the buffer becomes empty for the first time when the service of packet k is finished. It is obvious that all the events E k , k = 1, • • • N, are mutually exclusive. Then, the event of starvation is the union ∪ N -1 k=x1 E k . This union of events excludes E N because the empty buffer after the service of N packets is not a starvation. When the buffer is empty at the end of the service of the k th packet, the number of arrivals is kx 1 after the prefetching process. The probability of having kx 1 arrivals and k departures is computed from a binomial distribution,

( 2k-x1 k-x1 ) p k-x1 (1 -p) k .
We next find the necessary and sufficient condition of the event E k . If we have a backward time axis that starts from the time point when the buffer is empty for the first time, the number of departure packets is always more than that of arrival packets. As a result, the Ballot Theorem can be applied. For example, among the last m events (i.e. m ≤ 2kx 1 ), the number of packets that have been played is always greater than the number of arrivals. Otherwise, the empty buffer already happens before the k th packet is served.

According to the Ballot theorem, the probability of event E k is computed by x1 2k-x1

( 2k-x1 k-x1 ) p k-x1 q k .
Therefore, the probability of starvation, P s , is the probability of the union ∪ N -1 k=x1 E k , given by eq.( 1). The starvation event may happen for more than once during the file transfer. We are particularly interested in the probability distribution of starvations, given a finite file size N . The maximum number of starvations is J = ⌊ N x1 ⌋ where ⌊•⌋ is the floor of a real number. We define path as a complete sequence of packet arrivals and departures. The probability of a path depends on the number of starvations. We illustrate a typical path with j starvations in Figure 1. To carry out the analysis, we start from the event that the first starvation takes place. We denote by k l the l th departure of a packet that sees an empty queue. We notice that the path can be decomposed into three types of mutually exclusive events as follows:

• Event E(k 1 ): the buffer becoming empty for the first time in the entire path.

• Event S l (k l , k l+1 ): the empty buffer after the service of packet k l+1 given that the previous empty buffer happens at the departure of packet k l .

• Event U j (k j ): the last empty buffer observed after the departure of packet k j . November 30, 2013 DRAFT Obviously, a path with j starvations is composed of a succession of events

E(k 1 ), S 1 (k 1 , k 2 ), S 2 (k 2 , k 3 ), • • • , S j-2 (k j-2 , k j-1 ), S j-1 (k j-1 , k j ), U j (k j ).
We let P E(k1) , P Sl(kl,kl+1) and P Uj (kj) be the probabilities of events E(k 1 ), S l (k l , k l+1 ) and U j (k j )

respectively. The main difficulty to analyze the probability mass function is that the media player pauses for x 1 packets upon starvation. In what follows, we analyze the probabilities of these events step by step.

The event E(k 1 ) can happen after the departure of packet

k 1 ∈ [x 1 , N -1].
According to the proof of Theorem 1, the probability distribution of event E(k 1 ) can be expressed as

P E(k1) :=    0 if k 1 < x 1 or k 1 = N ; x1 2k1-x1 ( 2k1-x1 k1-x1 ) p k1-x1 q k1
otherwise .

(

) 2
The first starvation cannot happen at the departure of first (x 1 -1) packets because of the prefetching of x 1 packets. It cannot happen after all N packets have been served because this empty buffer is not a starvation. We next solve the probability distribution of the event U j (k j ). Suppose that there are j starvations after the service of packet k j . The extreme case is that these j starvations take place consecutively. Thus, k j should be greater than jx 1 -1. Otherwise there cannot have j starvations. The starvation event cannot take place after the departure of packet N because the whole file has now been transferred. If k j is no less than Nx 1 , the media player resumes until all the remaining Nk j packets are stored in the buffer. Then, starvation will not appear afterwards. In the remaining cases, the event U j (k j ) is equivalent to the event that no starvation happens after the service of packet k j . We can take the complement of starvation probability as the probability of no starvation. Hence, the probability distribution of event U j (k j ) is given by

P Uj(kj) :=                0, if k j < jx 1 or k j = N ; 1, if N -x 1 ≤ k j < N ; 1 - ∑ N -kj-1 m=x1 x1 2m-x1 ( 2m-x1 m ) p m-x1 q m , otherwise . (3)
We denote by P s (j) the probability of having j starvations. The probability P s (0) can be obtained from Theorem 1 directly. For the case with one starvation, P s (1) is solved by

P s (1) = N ∑ i=1 P E(i) P U1(i) = P E • P T U1 (4)
where T denote the transpose. Here, P E is the row vector of P E(i) , and P U1 is the row vector of P U1(i) ,

for i = 1, 2, • • • , N . November 30, 2013 DRAFT
To compute the probability of having more than one starvations, we need to find the probability of event S l (k l , k l+1 ) beforehand. Solving P Sl(kl,kl+1) is non-trivial due to that the probability of this event depends on the remaining file size and the number of starvations. After packet k l is served, the l th starvation is observed. It is clear that k l should not be less than lx 1 in order to have l starvations. Given that the buffer is empty after serving packet k l , the (l + 1) th starvation cannot happen at

k l+1 ∈ [k l + 1, k l + x 1 -1]
because of the subsequent prefetching process. Since there are j starvations in total, the (l+1) th starvation must satisfy k l+1 < N -(jl -1)x 1 . We next compute the remaining case that the l th and the (l + 1) th starvations happen after packets k l and k l+1 are served. Then, there are (k l+1k l ) departures, and

(k l+1 -k l -x 1
) arrivals after the prefetching process. According to the Ballot theorem, a path without starvation between the departure of packet (k l + 1) and that of packet (k l+1 ) is expressed as

x1 2kl+1-2kl-x1
. Therefore, we can express P Sl(kl,kl+1) as

         x1 2kl+1-2kl-x1 ( 2kl+1-2kl-x1 kl+1-kl-x1 ) p kl+1-kl-x1 q kl+1-kl , if k l ≥ lx 1 , k l + x 1 ≤ k l+1 < N -(j -l -1)x 1 ; 0, otherwise . (5)
We denote by P Sl the matrix of

P Sl(kl,kl+1) for k l , k l+1 ∈ [1, N ].
Here, P Sl is an upper triangle matrix where all the elements in the first (lx 1 -1) rows, and the last x 1 rows are 0. The probability of having j(j ≥ 2) starvations is given by

P s (j) = N ∑ k1=1 N ∑ k2=1 • • • N ∑ kj-1=1 N ∑ kj =1 P E(k1) • P S1(k1,k2) • • • P Sj-1(kj-1,kj ) • P Uj (kj ) = P E ( j-1 ∏ l=1 P S l ) P T Uj . (6)
The probability of no starvation, P s (0), is computed as 1-P s where P s is obtained from eq.( 1). Since the starvation event takes non-negative integer values, we can write the probability generating function (p.g.f.) G(z) by

G(z) = E(z j ) = ∑ J j=0 P s (j) • z j . (7)
In P, P Sl and P Uj , the binomial distributions can be approximated by the corresponding Normal distributions with negligible errors (see technical report [START_REF] Xu | Probabilistic Analysis of Buffer Starvation in Markovian Queues[END_REF]). The Gaussian approximation significantly reduces the computational complexity of binomial distributions.

We next analyze the complexity of matrix (including vector) operations in eq. ( 6). A matrix operation consists of floating-point operations where one floating-point operation can be an addition, subtraction, multiplication or division of two float type matrix elements [START_REF] Golub | Matrix Computations[END_REF]. In the complexity analysis of matrix operations, the lower-order terms are usually ignored. The approximated probability of starvation in eq.( 1) consists of N additions, thus having a complexity order O(N ). The probability of one starvation is a November 30, 2013 DRAFT product of two vectors, which consists of N multiplications and N -1 additions. Hence, the complexity order in still O(N ). If there are only two starvations, we need to compute the product of two vectors and one matrix, which has a complexity order O(N 2 ). When j ≥ 3, the computation of P s (j) involves the product of two matrices. In general, multiplying two matrices has a complexity order O(N 3 ). In eq.( 6), we should multiply a vector and a matrix each time, instead of multiplying two matrices inside the bracket. Then, the matrix operations in eq.( 6) only contain a set of multiplications between a vector and a matrix. This yields the complexity order O(N 2 ). To sum up, the computation of the p.g.f. of starvations in eq.( 7) has a complexity order O(N 2 ), given the start-up/rebuffering threshold x 1 and the file size N .

Asymptotic Property:

We want to know whether the starvation event yields simple implications as the file size N approaches ∞. The asymptotic behavior of the starvation probability is given by lim

N →∞ P s :=    1 if ρ < 1; exp ( x1(1-2p) 2pq ) otherwise . (8)
The detailed analysis can be found in the technical report [START_REF] Xu | Probabilistic Analysis of Buffer Starvation in Markovian Queues[END_REF].

The asymptotic starvation probability is irrelevant to the start-up threshold when ρ < 1. Under this situation, it is necessary to know how frequent the starvation event happens. Here, we compute the average time interval between two starvations. We let T s be the duration of starvation interval. Its expectation

E[T s ]
is the expected busy period of an M/M/1 queue with x 1 customers in the beginning [START_REF] Liu | Busy period in GI(X)/G/∞[END_REF], i.e.

E[T

s ] = x1 λ(1-ρ) .

C. Extension to Discrete-time Systems

In general, the playback rate of video streaming has a much smaller variance than the arrival rate.

Hence, the playback of streaming packets is sometimes regarded as a time-slotted process (e.g. [START_REF] Liang | Effect of delay and buffering on jitter-free streaming over random VBR channels[END_REF], [START_REF] Parandehgheibi | Avoiding Interruptions a QoE Reliability Function for Streaming Media Applications[END_REF])

where only one packet is served at the beginning of a time slot. We consider a playout buffer modeled as an M/D/1 queue. We denote by d the duration of a slot.

In this subsection, we introduce a discrete Ballot theorem named Takács Ballot Theorem.

Theorem 2: (Takács Ballot Theorem [1]) If X(1), X(2), • • • , X(l) are cyclically interchangeable r.v.s
taking on nonnegative integer values summing to k, then

P { t ∑ s=1 X(s) < t, ∀ t ∈ [1, l] } = [l -k] + l . (9)
November 30, 2013 DRAFT

The Takács Ballot Theorem presents a probability that the number of departures is larger than that of arrivals in all l slots. If the arrival process {X(s)} is Poisson, X(s) is i.i.d. at different slots, and thus cyclically interchangeable. Suppose that the starvation event happens after t packets have been served, (t ≥ x 1 ). The total number of arrivals is t-x 1 . We create a backward time axis where the starvation event happens at slot 1. The number of departures is always greater than that of arrivals. Otherwise, the starvation event has already taken place. Hence, according to Takács Ballot theorem, the probability of the departure always leading the arrival is

P { t ∑ s=1 X(s) < t, ∀t ∈ [1, l] } = x 1 l . (10)
Therefore, the probability that the first starvation takes place after the service of the l th packet (i.e.

starvaton event happening at slot (l+1))

P s (l) = x 1 l • P{ l ∑ s=1 X(s) = l-x 1 }, ∀l ≥ x 1 . (11)
For the Poisson process {X(s)}, the probability of l-x 1 packet arrivals in l slots (i.e. the duration ld) is obtained by

P{ l ∑ s=1 X(s) = l-x 1 } = (λld) l-x1 (l-x 1 )! exp(-λld). (12)
Given the file size N and the prefetching threshold x 1 , the starvation might happens upon the departure of packets from x 1 to N -1. Then the starvation probability is obtained by

P s = N -1 ∑ l=x1 x 1 l (λld) l-x1 (l-x 1 )! exp(-λld). (13)
We next show how the p.g.f. of starvation events can be derived using the Takács Ballot theorem. The path with j starvations is the same as that in Fig. 1. With certain abuse of notations, we reuse E(k 1 ), U j (k j ) and S l (k l , k l+1 ) to denote the first, the last and the other starvation events. According to eq.( 12), there has

P E(k1) :=    0 if k 1 < x 1 or k 1 = N ; x1 k1 (λk1d) k 1 -x 1 (k1-x1)! exp(-λk 1 d) otherwise . ( 14
)
Since there exist j starvations in total, the last starvation event will not happen at the departure of packets less than jx 1 . Given the last starvation happening as soon as the k th j packet is served, the probability of November 30, 2013 DRAFT no starvation afterwards can also be solved using eq.( 12).

P Uj(kj ) :=                0, if k j < jx 1 or k j = N ; 1, if N -x 1 ≤ k j < N ; 1- ∑ N -kj-1 s=x1 x1 s (λsd) s-x 1 (s-x1)! exp(-λsd), otherwise .
When the l th and the l+1 th starvation events appear at the departure of packet k l and k l+1 , the probability

P Sl(kl,kl+1) is given by          x1 kl+1-kl (λ(kl+1-kl)d) (k l+1 -k l -x 1 ) (kl+1-kl-x1)! exp(-λ(k l+1 -k l )d) if k l ≥ lx 1 , k l + x 1 ≤ k l+1 < N -(j -l -1)x 1 ; 0, otherwise . (15)
Then, the p.g.f. of starvation events can be solved using eq.( 7) in the same way.

IV. STARVATION ANALYSIS VIA A RECURSIVE APPROACH

In this section, we present a recursive approach to compute the p.g.f. of starvations based on [START_REF] Citon | Analysis of packet loss processes in high-speed networks[END_REF].

Compared with the one using Ballot theorem, the recursive approach can handle more complicated arrival process.

A. Probability of Starvation

The probability of starvation and the p.g.f can be analyzed all in once. However, we compute them separately because the analysis of the starvation probability provides an easier route to understand this approach.

We denote by P i (n) the probability of starvation with a file of n packets, given that there are i packets in the system just before the arrival epoch of the first packet of this file. In the original system, our purpose is to obtain the starvation probability of a file with the size N when x 1 packets are prefetched before the service begins. This corresponds to P i (n) with n = N +1-x 1 and i = x 1 -1. Here, the expression i = x 1 -1 means that the service starts when the x 1 -th packet sees x 1 -1 packets accumulated in the buffer. When the service begins, there are already x 1 packets in the queue. To compute P i (n), we will introduce recursive equations. We define a quantity

Q i (k), i = 0, 1, • • • , n, 0 ≤ k ≤ i, which is
the probability that k packets out of i leave the system during an inter-arrival period. This probability is equivalent to the probability of k Poisson arrivals with rate µ during an exponentially distributed period November 30, 2013 DRAFT with parameter 1/λ. According to [START_REF] Papoulis | Probability, Random Variables and Stochastic Processes[END_REF], we obtain

Q i (k) = ρ ( 1 1 + ρ ) k+1 = pq k , 0 ≤ k ≤ i -1, (16)
Q i (i) = ( 1 1 + ρ ) i = q i . ( 17
)
To carry out the recursive calculation, we start from the case n = 1.

P i (1) = 0, ∀i ≥ 1. (18)
When the file size is 1 and the only packet observes a non-empty queue, the probability of starvation is 0 obviously. If i is 0, the starvation happens for sure, thus yielding

P 0 (n) = 1, ∀n. (19)
For n ≥ 2, we have the following recursive equations:

P i (n) = i+1 ∑ k=0 Q i+1 (k)P i+1-k (n -1), 0 ≤ i ≤ N -1. (20)
We explain [START_REF] Alcock | Application flow control in YouTube video streams[END_REF] as the following. When the first packet of the file arrives and sees i packets in the system, the starvation does not happen. However, the starvation might happen in the service of remaining n -1 packets. Upon the arrival of the next packet, k packets out of i + 1 leave the system with probability

Q i+1 (k).
We next add constraints to the recursive equation [START_REF] Alcock | Application flow control in YouTube video streams[END_REF] for a file of size N . Since the total number of packets is N , the starvation probability must satisfy P i (n) = 0 for i + n > N .

B. P.G.F. of Starvations

To compute the p.g.f. of starvation, we use the same recursive approach, despite of the more complicated structure. With certain reuse of notation, we denote by P i (j, n) the probability of j starvation of a file with size n, given that the first packet of the file sees i packets in the system upon its arrival. Our final purpose is to compute the probability of starvation for a file of size N . It can be obtained from P i (j, n)

with i = x 1 -1 and n = N +1-x 1 .
When the first packet of remaining n packets arrives at the buffer, it sees x 1 -1 packets. At this time point, there are x 1 packets in the buffer and the service of packets begins.

In order to compute P i (j, n) recursively, we provide the initial conditions first:

P i (j, 1) =    0 ∀i = 1, 2, • • • , N -1, and j ≥ 1; 1 ∀i = 1, 2, • • • , N -1, and j = 0, (21)
November 30, 2013 DRAFT and

P 0 (j, 1) =    0 j = 0 or j ≥ 2; 1 j = 1. (22)
The equation [START_REF] Xu | Cross-Layer Qos Scheduling for Layered Multicast Streaming in OFDMA Wireless Networks[END_REF] means that the probability of no starvation is 1 conditioned by i ≥ 1 and n = 1.

Thus, the probability of having one or more starvations is 0 obviously if the only packet sees a nonempty system. The equation [START_REF] Xu | Probabilistic Analysis of Buffer Starvation in Markovian Queues[END_REF] reflects that the starvation happens for sure when the only packet observes an empty queue. However, there can only have one starvation event due to n = 1. Another practical constraint is

P i (j, n) = 0, if i + n > N (23)
because of the finite file size N .

To compute P i (j, n), we need to know what will happen if the buffer is empty, i.e. i = 0. One intuitive observation is

P 0 (0, n) = 0, ∀ 1 ≤ n ≤ N -b; (24)
where b := x 1 -1 is denoted to be the prefetching threshold. Eq.( 25) holds because an empty queue means at least one starvation event. For a more general probability P 0 (j, n), we begin with the case j = 1. If n ≤ b and the first packet of n sees an empty buffer, there has only one starvation, that is,

P 0 (1, n) = 1, ∀ 1 ≤ n ≤ b, (25)
If n > b, b packets will be prefetched. Thus, the remaining file size is nb. We see b packets in the system upon the arrival of the first packet in the remaining file. Given that the only one starvation event has taken place, there will be no future starvations. Therefore, the following equality holds,

P 0 (1, n) = P b (0, n -b), ∀ b < n ≤ N -b. (26)
Using the similar method, we can solve P 0 (j, n) for j > 1. However, the property of P 0 (j, n) with j > 1 is quite different

P 0 (j, n) = 0, ∀ j > 1 and 1 ≤ n ≤ b. ( 27
)
This means that the probability of having > 1 starvations is 0 if the file size is no larger than b. If n is greater than b, then b packets are prefetched, leaving nb packets in the remaining file. The remaining nb packets encounter j -1 starvations, given that the first packet sees b packets in the system upon arrival, i.e.

P 0 (j, n) = P b (j -1, n -b), ∀ j > 1 and n > b. (28)
November 30, 2013 DRAFT So far, we have computed a critical quantity P 0 (j, n), the probability of meeting an empty buffer. Next, we construct recursive equations to compute P i (j, n) as the following:

P i (j, n) = i+1 ∑ k=0 Q i+1 (k)P i+1-k (j, n -1) = i ∑ k=0 pq k P i+1-k (j, n -1) + q i+1 P 0 (j, n -1), (29)
for 0 ≤ i ≤ N -1. The eq.( 29) contains two parts. The former expression reflects the cases that the next arrival sees an non-empty queue. The latter one characterizes the transition of the system to a prefetching process that is computed by [START_REF] Krishnan | Video Stream Quality Impacts Viewer Behavior: Inferring Causality using Quasi-Experimental Designs[END_REF].

We are interested in how efficient the recursive method is. Hence, we present the roadmap to compute P i (j, n) and its complexity:

• Step 1: Solving P i (0, 2), for i = 1 to N -2;

• Step 2: Solving P i (0, n), for i = 1 to N -2, and n = 3 to N -x 1 +1 based on Step 1;

• Step 3: Adding j by 1 and computing P i (j, n) based on Step 1 and Step 2.

The complexity analysis is carried out from this roadmap. An operation in this recursive algorithm refers to an addition. In step 1, the computation of P i (0, 2) incurs i+2 additions for each i according to eq.( 29).

Hence, the total number of additions for all i from 1 to N is around N 2 /2.

Step 2 computes P i (0, n)

repeatedly for each n ∈ {2, • • • , N }. The Step 3 repeats
Step 1&2 for each j, but not augmenting the complexity order in N . Therefore, the total complexity has the order O(N 3 ). The recursive algorithm obtains the starvation probabilities for all j and n, (j ≤ J, n ≤ N ) and all initial start-up threshold i

(1 ≤ i ≤ N ).
Remark 1: We compare the complexity of the Ballot approach and the recursive approach. First, the standard Ballot approach contains factorial terms (e.g.

( 2k-x1 k-x1
) in eq.( 1)) that are of high computational burdens. Second, after Gaussian approximation of factorial terms, the Ballot approach has a complexity order O(N 2 ) given the file size and the start-up threshold. The recursive approach has a complexity O(N 3 ) for all combinations of initial start-up threshold i and file size n(1 ≤ i, n ≤ N ). Thus, the recursive approach has an overall smaller complexity than the Ballot approach.

C. ON/OFF Bursty Traffic

In this section, we model the arrival process as an interrupted Poisson process (IPP), which is commonly used to characterize the bursty and correlated arrivals. The source may stay for relatively long durations in ON and OFF states. The ON/OFF arrival model also has direct applications. For example, the Youtube servers use a simple ON/OFF rate control algorithm to transfer streaming packets November 30, 2013 DRAFT to the users [START_REF] Alcock | Application flow control in YouTube video streams[END_REF]. Our objective is to understand the interaction between the parameters of arrival process and the probability of starvation.

Fig. 2. Two-state Markov process to model bursty traffic

We illustrate the bursty traffic model in figure 2 with the state transition rates α and β. We denote by

Q i (k) ON , 0 ≤ i ≤ N -1, 0 ≤ k ≤ i
, the probability that k packets out of i leave the system upon an arrival at the ON state (i.e. no arrival during the OFF period). According to [START_REF] Citon | Analysis of packet loss processes in high-speed networks[END_REF], the following proposition holds.

Proposition 1: [START_REF] Citon | Analysis of packet loss processes in high-speed networks[END_REF] The probability Q i (k) ON is expressed as

Q i (k) ON = c 1 ( 1 a 1 ) k + c 2 ( 1 a 2 ) k , 0 ≤ k ≤ i -1, Q i (i) ON = c 1 (1/a 1 ) i 1 -1/a 1 + c 2 (1/a 2 ) i 1 -1/a 2 , (30)
where a 1 , a 2 , c 1 and c 2 are solved by

∆ = (λ + α + β) 2 -4λβ, a 1,2 = 1 + λ + α + β 2µ ± √ ∆ 2µ , c 1 = λ(β + µ) -λµa 1 a 1 (a 2 -a 1 ) , c 2 = λ(β + µ) -λµa 2 a 2 (a 1 -a 2 ) .
We next show how the starvation probability P i (j, n) is obtained. The starvation event can happen in both the ON and OFF states. However, the starvation event at the OFF state is equivalent to the event that the first new packet arrival at the ON state sees an empty queue. Therefore, we can use [START_REF] Dobrian | Understanding the Impact of Video Quality on User Engagement[END_REF] to compute the p.g.f. of starvations with bursty arrivals, simply replacing

Q i (k) by Q ON i (k). Remark 2:
The standard Ballot theorem cannot be used to study the starvations of the ON/OFF arrival process. In the presence of bursty traffic, the packet arrival process has two states, ON and OFF. The packet arrivals of two consecutive ON states are separated by an OFF state. Hence, the counting of the arrival arrivals is not equally probable.

V. FLUID MODEL ANALYSIS OF STARVATION PROBABILITY

So far we have studied the starvation behavior of a single file, which is concerned by either the media servers or the users. In fact, the streaming providers are more interested in the QoE evaluation scaled November 30, 2013 DRAFT to a large quantity of videos. They cannot afford the effort of configuring each file a different start-up delay. In this section, we present a fluid analysis of starvation probability, given the distribution of file size.

In the fluid model, the arrival and departure rates are deterministic. We let λ be the number of packet arrivals per second, and µ be the number of departures per second. Here, µ depends on the encoding rate that the media files use. We focus on the setting µ ≥ λ because no starvation will happen with µ < λ in the fluid model. We let x 1 be the start-up threshold. The start-up delay T 1 is simply computed by

x 1 /λ. Once the media packets are played, the queue length decreases at a rate µλ. The time needed to empty the queue is thus x1 µ-λ . We let N be the total number of packets that are served until a starvation happens,

N p = x 1 ( 1 + λ µ -λ ) = x 1 µ µ -λ . ( 31
)
If the file size is less than N p , there will be no starvation event.

The distribution of media file size depends on the types of contents. A measurement study in [START_REF] Cheng | Statistics and Social Network of YouTube Videos[END_REF] shows the distributions of Youtube video duration for four most popular categories: music, entertainment, comedy and sports videos. The authors find that most of the entertainment, comedy and sports videos are short. They are likely to follow exponential distribution or lognormal distribution with large standard deviations. The lengths of music files in playback time are usually between 180 and 240 seconds (the file size in Bytes is the product of duration and the default bit-rate on Youtube). Hence, with the help of the measurements in [START_REF] Cheng | Statistics and Social Network of YouTube Videos[END_REF], we speculate that music video files on Youtube follow lognormal distribution with a small standard deviation. The Pareto distribution adds practical restrictions to the video file size.

The file size needs to be greater than a certain value, and a very small fraction of video files can be very large. Today, Youtube allows some users to upload some long movies (more than the previous maximum of 10 min) without copyright issues. The distribution of movie file size may have a heavy-tail. Then, Pareto distribution can serve as a good approximation.

We compare the starvation probabilities of exponential, log-normal and Pareto distributions, given the start-up threshold. Note that these distributions possess the same mean file size. We assume that the users are homogeneous so that λ and µ are the same for different types of file size distributions.

i) Exponential distribution: Suppose that the file size N follows an exponential distribution with parameter θ. The probability of starvation, P

s , is obtained by

P (1) s = Prob (N > N p ) = exp(- θx 1 µ µ -λ ). ( 32
)
November 30, 2013 DRAFT ii) Pareto distribution: We let N m be the minimum possible value of the file size, and υ be the exponent in the Pareto distribution. The probability of starvation is computed by

P (2) s = Prob (N > N p ) =    ( Nm(µ-λ) µx1 ) υ ∀N m ≤ x1µ µ-λ ; 1 otherwise , (33)
where the expectation of the Pareto distribution is equal to that of the exponential distribution, i.e. where ϱ and σ are the mean and the standard deviation of a natural normal distribution. Given that N p packets can be served without an interruption, the starvation probability P (3) s is computed by

P (3) s = Prob (N > N p ) = 1 2 - 1 2 erf [ log x1µ µ-λ -ϱ √ 2σ ] , (34)
where its expectation exp(ϱ + σ 2 2 ) equals to 1 θ . Equations ( 32),( 33) and (34) show that the probability of starvation can be controled by setting x 1 , if the distribution of file size, the arrival and departure rates are pre-knowledge 1 .

VI. APPLICATION TO STREAMING SERVICE

This section presents four scenarios in streaming service in which our analysis can be utilized to optimize the objective QoE. Here, we focus on the M/M/1 system.

The QoE reflects the human perception of the streaming service. A common practice to evaluate QoE is called Mean Opinion Score (MOS). The video watchers give scores according to their subjective opinions.

The start-up delay and the starvation behaviors are explicitly defined as quality metrics related to user perception in [START_REF] Dobrian | Understanding the Impact of Video Quality on User Engagement[END_REF], [START_REF] Krishnan | Video Stream Quality Impacts Viewer Behavior: Inferring Causality using Quasi-Experimental Designs[END_REF] and [START_REF] Parandehgheibi | Avoiding Interruptions a QoE Reliability Function for Streaming Media Applications[END_REF]. To remove the confusion, we designate the direct human perception as subjective QoE, and designate the objective measure as objective QoE. The human tests implicitly map the objective QoE metrics into a single subjective value. This QoE value, though revealing the user perception statistically, is usually unreliable to report the QoE for each individual watching. At the same time, the subjective test is done after the watching, which cannot be utilized to tune the prefetching online. Therefore, a rising trend is to evaluate the objective QoE metrics and to balance the tradeoff among them (e.g. [START_REF] Dobrian | Understanding the Impact of Video Quality on User Engagement[END_REF], [START_REF] Krishnan | Video Stream Quality Impacts Viewer Behavior: Inferring Causality using Quasi-Experimental Designs[END_REF] and [START_REF] Parandehgheibi | Avoiding Interruptions a QoE Reliability Function for Streaming Media Applications[END_REF]). 1 Because the starvation probabilities P Our purpose is to use content prefetching as a way to achieve the optimal tradeoff between the start-up delay and the starvation behaviors (either the starvation probability or the continuous playback interval) for a user. In [START_REF] Parandehgheibi | Avoiding Interruptions a QoE Reliability Function for Streaming Media Applications[END_REF], the authors configure a start-up threshold to guarantee that the starvation probability is less than a certain value. The bound of the starvation probability is deemed as a parameter obtained from human tests. In this paper, we adopt a more flexible method by defining an objective QoE cost function for a user. A user-defined weight λ is introduced to indicate his/her preference to one type of objective QoE metrics. We first let the starvation probability be one of the QoE metrics. We let g(•) be a strictly increasing but convex function of the expected start-up delay E[T 1 ]. The larger the start-up delay, the higher the QoE cost. The convexity of g(•) means that streaming users are more and more impatient to large start-up delays. We denote by C 1 (x 1 ) the cost of a user watching the media stream,

C 1 (x 1 ) = P s + γg(E(T 1 )), (35)
where γ is a positive constant. A large γ represents that the users are more sensitive to the start-up delay, and a smaller γ means a higher sensitivity to the starvation. Our goal is to find the optimal start-up threshold x * 1 to minimize C 1 (x 1 ). The choice of C 1 (x 1 ) should satisfy three basic principles. First, it is convex in x 1 so that only one optimal threshold x * 1 exists. Second, C 1 (x 1 ) is bounded even if ρ is close to 1. Otherwise, the configuration of x 1 is extremely sensitive to ρ. Third, though x * 1 is not required to be a decreasing function of the arrival rate λ, it cannot grow unbounded when λ is large enough. In what follows, we simply let g(E(T 1 )) := (E(T 1 )) 2 = ( x1 λ ) 2 . Note that building a completely convincible cost function is very difficult for a particular user. Even the measurement studies in [START_REF] Dobrian | Understanding the Impact of Video Quality on User Engagement[END_REF] and [START_REF] Krishnan | Video Stream Quality Impacts Viewer Behavior: Inferring Causality using Quasi-Experimental Designs[END_REF] only quantify the influence of P s and E(T 1 ) statistically and indirectly (using user engagement). We choose C 1 (x 1 ) to be the sum of the measures of two types of objective QoE metrics for two reasons. One is the reflection of QoE tradeoff. The other is its simplicity. Other forms of C 1 (x 1 ) can be optimized in the same way. The more realistic cost function is subject to our future study.

We apply our models to optimize objective QoE in three scenarios: i) finite media streaming, ii) everlasting media streaming and iii) file level. The scenarios i) and ii) are designed for a single stream, while iii) is designed for a large number of streams. When the streaming file has a finite size, the congested bottlenecks such as the 3G base station or the wifi access point can configure or suggest a start-up threshold before the media stream is played. If the steaming file is large enough (e.g. realtime sport channel), a user can measure the arrival/service processes, and then configure the rebuffering threshold.

In the third scenario, the media server can set up one same start-up threshold for all the videos that November 30, 2013 DRAFT it distributes. To avoid malfunctions in realistic scenarios, a user can configure the maximum and the minimum start-up/rebuffering delay. Once maximum value is reached, the media player starts to play regardless of the prefetching threshold.

A. Finite Media Size

We hereby consider the adaptive buffering technique for a streaming with finite size. The eq.( 1) and eq.( 35) yield

C 1 (x 1 ) = N -1 ∑ k=x1 x 1 2k -x 1 ( 2k -x 1 k -x 1 ) p k-x1 (1 -p) k + γ( x 1 λ ) 2 . ( 36
)
The starvation probability decreases and the start-up delay increases strictly as x 1 grows. In the objective QoE optimization of finite media size, there does not exist a simple expression of the optimal threshold

x * 1 . To find x * 1 numerically, we need to compare the costs from all possible thresholds. The complexity order is low if the binomial distribution in eq.( 1) is replaced by the Gaussian distribution. If a user can tolerate up to 1 starvations, P s will be replaced by the probability (P s (0) + P s (1)) according to eq.( 4).

B. Infinite Media Size

We revisit the user perceived streaming quality in two scenarios: 1) ρ ≥ 1 and 2) ρ < 1.

Case 1: ρ ≥ 1. The starvation probability converges to a fixed value when the file size approaches infinity.

We adopt the same QoE metric as that of the finite media size. Note that P s can be directly replaced by its asymptotic value in eq.( 8). Submitting P s to C 1 (x 1 ), we have the following cost function

C 1 (x 1 ) = exp ( x 1 (1 -2p) 2pq ) + γ( x 1 λ ) 2 .
Letting the derivative dC1 dx1 be 0, we obtain

x 1 • exp ( x 1 (2p -1) 2pq ) = (2p -1)λ 2 4γpq .
The optimal threshold x * 1 is solved by

x * 1 = LambertW ( ( (2p -1)λ 2pq ) 2 • 1 2γ ) • 2pq 2p -1 ,
where LambertW (•) is the Lambert W-function.

Case 2: ρ < 1. When ρ < 1, P s is 1 for an infinite media size. If we adopt the QoE metric C 1 directly, the optimal start-up delay is always 0. This requires a new objective QoE metric for the case ρ < 1.

November 30, 2013 DRAFT Since the starvation happens many times, the continuous playback interval can serve as a measure of users' satisfaction. We denote by C 2 (x 1 ) the cost function for an infinite media size with ρ < 1,

C 2 (x 1 ) := exp(- δx 1 λ(1 -ρ) ) + γ( x 1 λ ) 2 ,
where δ is a user defined weighting factor to the expected playback duration (δ := 1 in our numerical examples). We differentiate C 2 (x 1 ) over x 1 , and let the derivative be 0, then the optimal start-up/rebuffering threshold is

x * 1 = LambertW ( δ 2 2γ(1 -ρ) 2 ) • λ(1 -ρ) δ .

C. Optimal Objective QoE in the File Level

Unlike the above QoE optimizations, the threshold x 1 for many files is configured by the media server, instead of the users. The objective is still to balance the tradeoff between the start-up delay and the starvation probability. Here, only the exponentially distributed file size is considered. We choose the cost

function C 1 (x 1 ) that yields C 1 (x 1 ) = exp(-θx1µ µ-λ ) + γ ( x1 λ ) 2 .
The optimal threshold x * 1 can be easily found as

x * 1 = LambertW ( ( θµλ µ -λ ) 2 • 1 2γ ) • µ -λ µθ .

VII. NUMERICAL EXAMPLES

A. Starvation of M/M/1 Queue

This set of experiments compares the probability of starvations with the event driven simulations using MATLAB. The M/M/1 queue is tested for up to 5000 times with arrivals from files of different sizes.

We deliberately consider four combinations of parameters: ρ = 0.95 or 1.1, and x 1 = 20 or 40 pkts.

The departure rate µ is normalized as 1 if not mentioned explicitly. The choice of the start-up thresholds coincides with the playout of audio or video streaming services in roughly a couple of seconds (e.g.

200∼400kbps playback rate on average given the packet size of 1460 bytes in TCP). The file size in the experiments ranges between 40 and 1000 in terms of packets. Figure 3 displays the probability of 0,1, and 2 starvations with parameters ρ = 0.95 and x 1 = 20. When the file size grows, the probability of no starvation decreases. We observe that the probabilities of 1 and 2 starvations increase first, and then decline after reaching the maximum values. The reason lies in that the traffic intensity ρ is less than 1. starvation is improved by more than 10% (e.g. N ≥ 300) when x 1 increases from 20 to 40. Figure 5 also validates the asymptotic probability of no starvation obtained from Gaussian and Riemann integral approximations etc. Figure 6 plots the probability of one starvation with the same parameters. Recall that the probability of one starvation decreases to 0 as N increases in the case ρ = 0.95. While figure 6 exhibits a different trend along with the increase of file size. This probability becomes saturated, instead of decreasing to 0. When ρ is greater than 1, the probability of having a particular number of starvations approaches a constant. In both figure 5 and6, simulation results validate the correctness of our analysis.

Hence, in the following experiments, we only illustrate the analytical results.

B. Starvation of Bursty Traffic

We consider the ON/OFF bursty arrival of packets into an M/M/1 queue. For the ease of comparison, we let the transition rates α and β be both 0.2. The file size ranges from 40 to 500 pkts. In figure 7, we November 30, 2013 DRAFT plot the probabilities of having no more than two starvations with ρ = 1.5 and x 1 = 40. As the file size increases, the probability of no starvation decreases. The probabilities of 1 and 2 starvations increases first, and then decreases to 0. This means that the starvation is for sure when the file size approaches infinity. In figure 8, we plot the starvation probabilities for ρ = 2.5 and 3.0 where the start-up threshold is set to 20. In contrast to figure 7, the probability of no starvation converges to a positive constant as N is large enough. In figure 10, we plot the probability of no starvation with ρ = 2.5 and N = 800 pkts. The transition rates α and β increases from 0.05 to 0.25. It can be seen that the probability of no starvation increases monotonically with the symmetric transition rates α and β.

C. Starvation in the File Level

This set of numerical experiments shows the relationship between the starvation probability and the distribution of file size. The traffic intensity ρ is set to 0.95. We let θ be 1/2000 in the exponential distribution. Then, the average file size is 2000 pkts. This setting is in accord with the recent measurement that most of mobile streaming files are short (i.e. a median size of 1.68MBytes) [START_REF] Liu | A Servers Perspective of Internet Streaming Deliver to Mobile Devices[END_REF]. For the Pareto distribution, we set the minimum file size to be 300 pkts so that the exponent υ is 1.1765. The parameters of the Log-normal distribution are set to (ϱ, σ) = (7.476, 0.5) and (7.101, 1.0) respectively. We plot the CDF curves of the file sizes in Figure 11. In this setting, the Pareto distribution exhibits an obvious heavy tail property. For the log-normal distribution, more than 30% percent of files are less than 1000 packets with σ = 1 and most of the files are less than 2000 packets with σ = 0.5. The exponential distribution and the log-normal distribution with σ = 1.0 exhibit similar CDF of file sizes. We evaluation the starvation probabilities in Figure 12 by increasing the start-up threshold from 20 to 200. The starvation probability of the Pareto distributed files has a sharp reduction in the beginning of increasing x 1 . However, as x 1 is more than 80, there is only a slight improvement in the starvation probability. Hence, the streaming service providers need to configure different start-up thresholds for the short files and the tail files in the Pareto distribution. For the log-normal distribution with σ = 0.5, the starvation probability is high with small x 1 . There have significant reductions of starvation probability when x 1 increases from 40 to 140. This is because the file sizes have a small standard deviation. Very small thresholds do not help to reduce the starvation probability and very large thresholds do not further reduce the starvation probability. When November 30, 2013 DRAFT

x 1 increases from 20 to 200, we can observe the noticeable reduction of starvation probability in the exponential file size distribution. As the take-home message of fluid analysis, the choice of x 1 relies on the distribution of file size to a great extent. To obtain a better objective QoE, the media service providers can set different x 1 for different categories of media files.

D. Optimizing Quality of Experience

Objective QoE optimization of finite media size:

We illustrate the total QoE cost (including the starvation cost and the start-up delay cost) in figure 13 with λ = 16, 20, 24 and µ = 25. The file size is set to N = 1000 and the weight γ is 10 -3 . We find that the total QoE looks neither "concave" nor "convex" w.r.t. the start-up threshold. For example, when x 1 is less than 300 with λ = 16, the increase in the start-up delay cost cannot be compensated by the reduction of the starvation probability. We further plot the optimal start-up threshold obtained from the maximum of eq.( 36) in figure 14. When γ = 10 -4 and γ = 10 -3 , the optimal start-up threshold x * 1 decreases when λ increases. We also observe that for each λ x * 1 of the case γ = 10 -4 is higher than that of γ = 10 -3 because the former user is more sensitive to the starvation. In the extreme scenario γ = 0, the streaming user will download the whole media file before watching it. In the case γ = 5 × 10 -3 , x * 1 are always 1 if the arrival rate λ is less than 20. The reason lies in that the total cost is always greater than 1 in those situations. The start-up threshold x * 1 = 1 will induce numerous consecutive starvations, which definitely degrades the streaming QoE. To mitigate this malfunction, we can introduce a minimum playback delay that works independent of the QoE optimization. We plot the optimal prefetching thresholds x 1 for the case ρ > 1 in figure 15 and the case ρ < 1 in figure [START_REF] He | New Analysis Framework for Discrete Time Queueing Systems with General Stochastic Sources[END_REF]. As λ increases, the optimal prefetching threshold x * 1 reduces. Unlike figure 14, there do not exist an abrupt change in x * 1 . This is because the cost function C 1 (x 1 ) is a convex function of x 1 with both ρ > 1 and ρ < 1. Furthermore, x * 1 decreases as γ increases (the user putting more weight to the prefetching delay). We investigate the cost minimization problem at the media server side numerically. We let µ := 25 which means that 25 packets are served per second. Given the packet size of 1460 bytes, this service rate is equivalent to 292Kbps (without considering protocol overheads). We let the mean file size 1/θ be 1000 and 2000 packets respectively (equivalent to the playback time of 40 and 80 seconds). The sensitivity γ is set to 0.01 or 0.005. Figure 17 illustrates the choice of the optimal start-up thresholds when λ increases from 20 to 25 (i.e. ρ ≤ 1). We evaluate four combinations of θ and γ numerically. Our observations are summarized as follows. First, for the same file size distribution, a smaller γ causes a higher optimal start-up threshold. Second, x * 1 is not a strictly decreasing function of λ. When λ is small (e.g. 20pkts/s), a large start-up threshold does not help much in reducing the starvation probability, but causes impatience of users of waiting the end of prefetching. If λ increases, the adverse impact of setting a larger x 1 on the start-up delay can be compensated by the gain in the reduction of starvation probability. Third, with the same sensitivity γ, the optimal x * 1 of a long video stream can be smaller than that of a short one in some situations. This is caused by the fact that the large threshold might not significantly improve the starvation probability for a file of large size. We further show the starvation probability in figure 18.

A larger mean file size, or a smaller γ result in a larger probability of starvation. Unlike the start-up threshold x * 1 , the starvation probability is shown to be strictly decreasing as λ increases.

November 30, 2013 DRAFT

VIII. CONCLUSION AND DISCUSSION

We have conducted an exact analysis of the starvation behavior in Markovian queues with a finite number of packet arrivals. We perform a packet level analysis and a fluid level analysis. The packet level study is carried out via two approaches, the Ballot theorem and the recursive equations. The former provides an explicit expression, but is usually limited to i.i.d. packet arrivals. The latter can handle bursty packet arrival process, but without an explicit result. From the perspective of a media service provider, we perform a fluid level analysis that computes the probability of starvation for a large number of video files. We further apply the theoretical results to tune the prefetching thresholds in order to optimize the objective QoE for media streaming services.

Fig. 1 .

 1 Fig. 1. A path with j starvations

υNm υ- 1

 1 = 1 θ .iii) Log-Normal distribution: We suppose that the file size follows a log-normal distribution ln N (ϱ, σ),

  complicated forms, we will compare their dependency on x1 numerically in section VII. Both Pareto and Log-normal distributions have two parameters. In the comparison, we fix one of them, and solve the other according to the property of identical expectations.November 30, 2013 DRAFT

Figure 3 Figure 5

 35 Figure 3 also shows that our analytical results match the simulation well.Figure 4 exhibits the similar results when the start-up threshold is 40 pkts. The comparison between figure 3 and 4 manifests that a larger x 1 is very effective in reducing starvation probability.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Probability of 0, 1, and 2 starvations with ρ = 0.95 and x1 = 20

Fig. 5 .Fig. 6 .November

 56 Fig. 5. Probability of no starvation with ρ = 1.1: x1 = 20 and x1 = 40

Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .

 789101112 Fig. 7. ON/OFF traffic: probability of 0, 1, and 2 starvations with ρ = 1.5 and x1 = 40

Figure 9

 9 Figure 9 illustrates the impact of x 1 on the probability of no starvation. In this set of experiments, ρ is set to 2. The start-up threshold x 1 increases from 20 to 60 pkts, and the file size increases from 400 to 800 pkts. It is clearly shown that a slight increase in x 1 can greatly improve the starvation probability.

24 Fig. 13 .

 2413 Fig. 13. Finite media size: total cost with µ = 25 and γ = 0.001

Fig. 14 .

 14 Fig. 14. Finite media size: optimal thresholds with γ = 10 -4 , 10 -3 , and 5 × 10 -3

Fig. 15 .Fig. 16 .

 1516 Fig. 15. Optimal threshold x * 1 of infinite file size: ρ > 1

  ; γ=0.01 1/θ=1000; γ=0.005 1/θ=2000; γ=0.01 1/θ=2000; γ=0.005

Fig. 17 .

 17 Fig. 17. Optimal threshold x * 1 for QoE enhancement at the file level: µ = 25

November 30, 2013 DRAFT


I Stefan

F De Oliveira

Santos

O Sorlin

T Davinson

M Lewitowicz

G Dumitru

J C Angélique

M Angélique

E Berthoumieux

C Borcea

R Borcea

A Buta

J M Daugas

F De Grancey

M Fadil

S Grévy

J Kiener

A Lefebvre-Schuhl

M Lenhardt

J Mrazek

F Negoita

D Pantelica

M G Pellegriti

L Perrot

M Ploszajczak

O Roig

M G Saint

I Ray

M Stanoiu

C Stodel

V Tatischeff

J C Thomas

Probing Nuclear forces beyond the drip-line using the mirror nuclei 16 N and 16 F

Keywords: numbers: 25.60.-t, 97.10.Cv, 25.70.Ef, 25.40.Cm, 21.10.-k, 27.20.+n

Radioactive beams of 14 O and 15 O were used to populate the resonant states 1/2 + , 5/2 + and 0 -, 1 -, 2 -in the unbound 15 F and 16 F nuclei respectively by means of proton elastic scattering reactions in inverse kinematics. Based on their large proton spectroscopic factor values, the resonant states in 16 F can be viewed as a core of 14 O plus a proton in the 2s 1/2 or 1d 5/2 shell and a neutron in 1p 1/2 . Experimental energies were used to derive the strength of the 2s 1/2 -1p 1/2 and 1d 5/2 -1p 1/2 proton-neutron interactions. It is found that the former changes by 40% compared with the mirror nucleus 16 N, and the second by 10%. This apparent symmetry breaking of the nuclear force between mirror nuclei finds explanation in the role of the large coupling to the continuum for the states built on an ℓ = 0 proton configuration.

I. INTRODUCTION

Theoretical description of particle-unbound nuclei in the framework of open quantum systems is a challenge to basic nuclear research [1]. In such systems, the coupling to the scattering continuum may lead to the modification of the effective interactions [2] and a further reordering of the shells [3]. The identification and understanding of the role of specific parts of the nuclear forces [3] in stabilising atomic nuclei and inducing shell evolutions is a central theme of nuclear physics [4]. This understanding would bring a better predictive power for exotic nuclei such as those involved in the explosive r-process nucleosynthesis or X-ray bursters where relevant spectroscopic information is not yet available [5]. A good way to shed light on the effect of the continuum is to compare level schemes of mirror nuclei involving a bound and an unbound nucleus. The asymmetries observed between the mirror nuclei, the so-called Thomas-Ehrman shifts [6], can be used to single out the role of nucleon-nucleon interaction.

An ideal case to study the effect of the continuum can be found in the mirror systems: unbound 16 9 F 7 and the bound 16 7 N 9 nuclei [7]. In the present work, we propose to determine the energies and widths of the unbound states in 15 F and 16 F using the resonant elastic scattering technique. The new measurements have the advantage to combine excellent energy resolution, high statistics and precise energy calibration that marks a leap in quality and consistency over the precedent results obtained for these nuclei. We shall use these properties to derive the proton-neutron interaction energies in the unbound 16 F nucleus and to determine the role of the continuum in changing effective interactions in mirror nuclei.

II. EXPERIMENT

The nuclei 16 F and 15 F were studied through the measurement of elastic scattering excitation functions. Elastic scattering reactions at low energies can be described by the Rutherford scattering process, but the excitation function also shows "anomalies", i.e. resonances that are related to states in the compound nucleus. The analysis of these resonances through the R-Matrix formalism [8] can be used to determine spectroscopic properties of the states, i.e. excitation energies, widths, and spins, see examples of analysis in Ref. [9,10]. Four different beams were used in this experiment: the two radioactive beams of 15 O and 14 O for the study of 16 F and 15 F, and the two stable beams 14 N and 15 N for the calibrations.

Radioactive 15 O 1+ ions were produced at the SPIRAL facility at GANIL through the fragmentation of a 95 AMeV 16 O primary beam impinging on a thick carbon production target. They were post-accelerated by means of the CIME cyclotron up to the energy of 1.2 AMeV, the lowest energy available at this accelerator. Intense (several nAe) stable 15 N 1+ and molecular ( 14 N 16 O) 2+ beams came along with the radioactive beam of interest. The selection of one of these species was obtained by using a vertical betatron selection device [START_REF] Bertrand | 17th International Conf. on Cyclotrons and their applications[END_REF] located inside the cyclotron, and by choosing the suitable magnetic rigidity of the LISE spectrometer [START_REF] Anne | [END_REF] after the nuclei have traversed a 38 µg/cm 2 carbon stripper foil located at the object focal point of LISE. It was possible to obtain an 15 O 6+ beam with an intensity of 1.0(2)x10 6 pps and 97(1) % purity, or one of two stable beams of 15 N 6+ or 14 N 6+ with 10 8 pps and 100 % purity.

The selected ions were sent onto a thick polypropylene (CH 2 ) n target in which they were stopped. Some ions underwent proton elastic scattering and the scattered protons were detected promptly to the reaction in a E(300 µm) silicon detector that covered an angular acceptance of ±1 • downstream the target. Resonances in the compound nucleus 16 F were studied through the analysis of the scattered protons spectrum obtained with the 15 O beam. Stable beams of 14 N and 15 N were used for calibration purposes to measure the elastic scattering reactions 1 H( 14 N,p) 14 N and 1 H( 15 N,p) 15 N in the same experimental conditions. Energy calibrations and resolutions were measured by populating known resonances in the compound nucleus 15 O [14]: the state J π =3/2 + , E R = 987 (10) keV, Γ = 3.6 (7) keV, and in the compound nucleus 16 O: J π =0 -, E R = 668(4) keV, Γ = 40(4) keV and J π =2 -, E R = 841(4) keV, Γ = 1.34(4) keV.

In the same manner, radioactive 14 O ions were produced with the intensity of 1.9(1)x10 5 pps and postaccelerated to 6 AMeV. The isobaric contamination of the beam was reduced down to 0.0(1) %. Scattered protons were detected in ∆E(500 µm)-E(6 mm cooled SiLi) silicon detectors that covered an angular acceptance of ±2.16 • . Resonances in the 15 F compound nucleus were studied through the analysis of the proton spectrum. A pure 14 N 6+ beam accelerated to 6 AMeV was used for calibrations.

The energy resolution of the measured scattered protons can be determined by the relation:

σ Lab = σ 2 det + σ 2 θ + σ 2 strag
, where σ det is the energy resolution of the detector that is σ det =9 keV (20 keV) in the 15 O ( 14 N) setting, σ strag is the energy straggling in the target that is estimated to be lower than 5 keV from simulations, and σ θ is the energy resolution due to the aperture dθ of the detector. In inverse kinematics it can be derived that σ θ = tan(θ)Edθ. Therefore the degradation in energy resolution is minimal when θ = 0 • . For this reason, and for maximizing the ratio between the nuclear and the Coulomb contribution of the differential cross-section, the scattered protons were measured at forward angles. An energy resolution of σ Lab = 10 keV was measured in the case of 16 F, which leads to σ CM ≃ 3 keV in the center of mass, and σ Lab = 22 keV or σ CM ≃ 7 keV in the case of 15 F.

III. RESULTS

The measured scattered protons spectra of 1 H( 15 O,p) 15 O and 1 H( 14 O,p) 14 O were transformed into the center of mass excitation functions by taking into account the energy losses, the energy and angular straggling, the intrinsic energy resolution and the angular acceptance of the Si detectors. This procedure was successfully tested using the 1 H( 14 N,p) 14 N and 1 H( 15 N,p) 15 N reactions (see the upper part of Fig. 1). Reactions with the carbon nuclei of the target, although very weak, were subtracted in the case of 15 F using data from Ref. [13], and neglected in the other systems. The incident energies in CM are 1.15 MeV and 5.6 MeV respectively. As the lowest excited states in 15 O and 14 O lie above 5.1 MeV, inelastic scattering do not contribute to the reaction with 14 O, and is expected to be negligible in the case of 15 O.

The spectra corresponding to 16 F and 15 F are shown in the lower part of Fig. 1. They were fitted using the R-matrix formalism with the code Anarki [8]. The shape and height of the peaks are used to derive the energy, J π and width of the resonances. The uniqueness of the solution was controlled carefully. Two broad resonances are found at S p =-1.31(1) MeV, Γ R =853(146) keV, J π =1/2 + and E R =2.78(1) MeV, Γ R =311(10) keV, J π =5/2 + in 15 F. Energies and widths of these resonances are consistent with previous results having larger uncertainties [16]. Three resonances corresponding to the 0 -, 1 -and 2 -states in 16 F were identified. Their energy and width are given in Table I. A proton separation energy S p = -534 ± 5 keV was obtained, in agreement with the value S p = -536 ± 8 keV recommended both in the compilation [14] and the recent measurement S p = -535 keV [13]. Conversely, the observed width of the 1 -resonance, 70 ± 5 keV, differs significantly with the recommended value of < 40 keV or with the recent experimental value of 103 ± 12 keV [13], and is in good agreement with the most recent measurement of 87 ± 16 keV [15]. New recommended weighted mean values are proposed in Table I for the three resonances, while the properties of the J π =3 - state are taken from Ref. [13]. The measured widths Γ exp p of resonances are related to the spectroscopic factors C 2 S through the relation Γ exp = C 2 S Γ sp , where Γ sp are the calculated single-particle widths [13]. As shown in Table I, the measured spectroscopic factors of the lowlying states in 16 F are all close to 1. It is also the case in the mirror nucleus 16 N [19,21].

A recent publication [20] reported very precise predictions of the 16 F low-lying states widths using charge symmetry of strong interaction and Woods-Saxon potentials. A 30 % disagreement is observed between these predictions and our very accurate experimental results. It suggests that a larger value of the radius should be used in the potential. As shown in Fig. 6 of Ref. [20], a value of 3.6 fm gives a much better agreement. 16 O, 16 F and 15 F can be observed. The lines are results of the R-matrix calculations using the parameters from Ref. [14] for the 15 O and 16 O nuclei, from Table I for the 16 F nucleus, and from this work for the two resonances observed in 15 F. TABLE I: Measured energies, widths and deduced spectroscopic factors C 2 S for the low-lying states in 16 F. The new recommended separation energy is Sp = -535 ± 5 keV. The spectroscopic factors are calculated using the method proposed in Ref [13].

Compilation ref [14] Ref [13] Ref [15] This

IV. INTERPRETATION

The comparison of the two level schemes of the mirror nuclei 16 N and 16 F is shown in Fig. 2. Large differences can be observed: the ground state of 16 F has 0 -while that of 16 N has 2 -, and the two 0 -and 1 -states are down shifted in energy relatively to the other states by more than 500 keV.

In these odd-odd nuclei, a good approach is to use a core plus two nucleons model. The first excited states of the 16 7 N 9 nucleus can be well described using a singleparticle description with a closed core of 14 C plus a deeply bound proton in the 1p 1/2 orbital (S p ( 15 N)=+10.2 MeV) plus a neutron in the 2s 1/2 orbital (S n ( 15 C)=+1.22 MeV), leading to J π =0 -, 1 -states, or plus a neutron in the 1d 5/2 orbital (S n ( 15 C*)=+0.48 MeV), lead-ing to J π =2 -, 3 -states (see Fig. 2). In the same way, (0,1) -and (2,3) -states in 16 F can be described as a 14 O core plus a neutron in the 1p 1/2 orbital (S n ( 15 0)=+13.22 MeV) plus a proton in the 2s 1/2 orbital (S p ( 15 F)=-1.31 MeV) or plus a proton in the 1d 5/2 orbital (S p ( 15 F*)=-2.78 MeV). This simplified single particle view is justified as the spectroscopic factor values of the systems ( 15 N = 14 C + p), ( 15 F = 14 O + p) and ( 15 C= 14 C + n) are close to unity [14,16,17]. In this framework, the experimental neutron-proton (n-p) effective interactions elements in 16 N, labeled Int exp 16 N (J), can be extracted from the experimental binding energies (BE) as in Ref. [18]:

Int exp
16 N (J) = BE( 16 N) J -BE( 16 N free ). In this expression BE( 16 N f ree ) corresponds to the bind-FIG. 2: Schematic view of the valence proton (π) and neutron (ν) orbits involved in the mirror systems 16 9 F7 and 16 7 N9 on top of the 14 8 O6 and 14 6 C8 core nuclei. The major difference between these two systems is that the proton 2s 1/2 or 1d 5/2 (not shown) orbits are unbound in 16 9 F7. The level schemes of the two mirror nuclei differ significantly, as shown in the bottom part of the figure .   ing energy of the 14 C+1p+1n system without residual interaction between the proton and the neutron. In the case of the J π =0 -and 1 -states:

BE( 16 N free ) = BE( 14 C) 0 + + BE(π1p 1/2 ) + BE(ν2s 1/2 ) where BE(π1p 1/2 ) = BE( 15 N) 1/2 --BE( 14 C) 0 + BE(ν2s 1/2 ) = BE( 15 C) 1/2 + -BE( 14 C) 0 +
Combining these equations, we obtain:

BE( 16 N free ) = BE( 15 C) 1/2 + + BE( 15 N) 1/2 --BE( 14 C) 0 +
The same method has been also applied to obtain the J π =2 -, 3 -states originating from the n-p coupling π1p 1/2 ⊗ ν1d 5/2 . The obtained experimental n-p interaction energies Int exp 16 N and Int exp 16 F are given in Table II. While the effective interactions of the J π =0 -,1 -states differ by as much as 40% in the mirror systems, the interaction energies of the J π =2 -,3 -states differ only by 10%, despite the fact that the proton 1d 5/2 orbit is less bound than the 2s 1/2 one by about 1.5 MeV.

Hereafter, we show that these changes of n-p effective interactions can be explained by the effect of the coupling with continuum which leads both to a change in the spatial overlap of the neutron and proton wave functions and in the Coulomb electrostatic energy. Since the n-p effective interaction energies (0.5-2 MeV) are much smaller than the mean nuclear potential energy (≈ 50 MeV), we use the approximation that proton and neutron radial wave functions u p (r, J) and u n (r, J) can be calculated by neglecting the n-p interaction, and by solving the are based on the interactions derived for 16 N to which the effects of the change in wave functions overlaps (over) between mirror nuclei and the change in Coulomb (over+C) energy have been added (see text for details).

J π = 0 -, 1 -[1p 1/2 ⊗ 2s 1/2 ] and J π = 2 -, 3 -[1p 1/2 ⊗ 1d 5/2 ]

State (J)

Int exp Schrödinger equation in Woods-Saxon nuclear potentials whose depths have been adjusted to reproduce the observed neutron or proton binding energies for the states in 16 N and 16 F. Results as shown Fig. 3. Wave functions are quasi-identical between 16 F (full lines) and 16 N (dashed lines) except for the 2s 1/2 wave functions. Using a schematic zero-range v pn J = a J δ(r p -r n ) interaction, the n-p interaction Int over (J) is [START_REF] Heyde | The Nucler Shell Model[END_REF]:

Int over (J) = a J 4π ∞ 0 1 r 2 [u p (r, J)u n (r, J)] 2 dr (1)
where a J contains the strength of the n-p nuclear interaction. As the zero-range delta function is only a crude approximation of the nuclear force, the a J coefficients have been adjusted to equate the calculated and experimental (Int exp 16 N (J)) interaction energies in 16 N. By virtue of the charge symmetry of nuclear forces, the same a J coefficients should be used to calculate the interaction energies r(fm) FIG. 3: (colors online) Calculated 1p 1/2 , 1d 5/2 and 2s 1/2 single particle radial wave functions un(r) and up(r) (see text) for 16 N (dashed lines) and 16 F (full lines). For the scattering states, the unbound proton wave function in the region between r=0 and r=16 fm is renormalized to 1 in order to have the proton inside the nucleus before it decays. The 2s 1/2 wave function of the unbound proton in 16 F is more spatially spread than it is for the bound neutron in 16 N, contrary to the 1d 5/2 unbound proton wave function that is retained inside the nucleus by the centrifugal barrier.

Int over 16 F (J) in 16 F. Comparison between experimental and calculated interaction energies given in Table II deserve several important remarks. Firstly, the Int over 16 F (2,3) values are similar to those in the mirror system Int exp 16 N (2,3). This can be explained by the fact that the unbound proton and bound neutron 1d 5/2 wave functions are similar in the mirror systems (see Fig. 3). This is due to the high ℓ=2 centrifugal barrier of ≈ 3 MeV and Coulomb barrier of ≈ 3 MeV that prevent the unbound proton in 16 F to couple strongly with the continuum. This is also confirmed with the very narrow measured widths of 5 keV and 15 keV of the 2 -and 3 -states. Secondly, the calculated Int over 16 F (0,1) values, built on the 2s 1/2 ⊗ 1p 1/2 coupling, are intermediate between the experimental values of 16 N and 16 F. As ℓ=0 protons (from 2s 1/2 ) in 16 F do not encounter any centrifugal barrier, their radial wave function is more extended, as shown in Fig. 3. Consequently the overlap between the wave functions of the 2s 1/2 unbound proton and the deeply bound 1p 1/2 neutron is reduced. Thirdly, as the Int over 16 F (J) values differ from the experimental Int exp 16 F (J) values, an additional effect is required to understand the differences.

While the change in Coulomb energy between two isotopes can be usually neglected for bound states, the apparent radial extension of the wave function of an unbound state can be larger than that of a bound state, leading to a change in Coulomb electrostatic energy E c (J). In the case of 16 F, the interaction energy should therefore be rewritten for each state J. As for the J=0,1 states it writes:

Int over+C 16 F (J) = Int over 16 F (J) + E c ( 16 F(J)) -E c ( 15 F) 1/2 +
Coulomb energies are determined by using the following relation between the charge distribution ρ(r) of the core 14 O nucleus with Z core =8 and a single proton in the 2s 1/2 or 1d 5/2 orbit having radial wave functions u p (r, J):

E c (J) = Z core 4πǫ 0 ∞ 0 ρ(r)u p (r, J) 2 r dr (2)
To give an example, it is found that E c ( 16 F (0 -)) -E c ( 15 F ) 1/2 + = 168 keV . By applying this Coulomb energy correction as well as the one due to the change in wave functions overlaps between mirror nuclei, it is found that the calculated Int over+C 16 F

(J) and experimental Int exp 16 F (J) interaction energies are very similar, as shown in Table II. This suggests that differences in the protonneutron interaction energies between the two mirror nuclei are very well accounted for by these two combined effects, the amplitude of which sensitively depends on the energy and angular momentum of the states under study.

Our results agree very well with those obtained by Ogawa et al. [7] where the mirror system 16 F-16 N was studied using different model which is based on one particle plus one hole on top of the 16 O inert core, the residual interaction being calculated with the M3Y interaction and single-particle wave functions obtained under the Woods-Saxon plus Coulomb potential. Compared to this work, our model is even simpler and explains perfectly the observed differences between the two mirror nuclei. This good agreement and the simplicity of our model makes this system a particularly interesting textbook case for understanding the effect coupling to continuum on effective nuclear forces and subsequent shell reordering.

As a warning, we would like to add that in our simplified approach, the effects of nuclear correlations to derive the effective nuclear forces in the A=16 mirror nuclei have been neglected. We consider that this assumption is reasonable due to the fact that correlations should be at the first order similar in the two mirror nuclei. We expect that the implementation of nuclear correlations will slightly change the intensity of the derived effective interactions, but this change would be similar in the two mirror nuclei. The role of correlations certainly deserves a dedicated treatment that goes beyond the scope of the present work.

V. CONCLUSION

The 1/2 + , 5/2 + and 0 -, 1 -, 2 -resonant states were studied in the 15 F and 16 F nuclei, respectively, with unprecedented energy accuracy and resolution by means of proton elastic scattering reactions in inverse kinematics. Experimental energies were used to deduce that the effective proton-neutron interactions between the mirror nuclei differ by as much as 40% for the J π =0 -,1 -states and by only 10% for the J π =2 -,3 -states, although the latter states lie at higher energy in the continuum. We demonstrate that these features are well explained by the effect of the presence of the proton wave functions in the continuum that reduces proton and neutron radial overlaps and induces significant changes in Coulomb energy. This mirror system is ideal to test models in which the role of the continuum is or will be implemented, such as in ab initio no-core shell model [START_REF] Navràtil | [END_REF], coupled cluster approach [24], or shell model approaches [1,25]. A correct treatment of continuum is required for predicting the location of drip-lines, modeling weakly bound nuclei such as halo nuclei and predicting resonant states of astrophysical importance for X-ray bursters and for the r-process nucleosynthesis.

2 FIG. 1 :

 21 FIG.1: Measured excitation functions of proton resonant elastic scattering on the 14 N, 15 N,15 O and 14 O nuclei. The differential cross sections measured at 180 • in the CM are shown as a function of the laboratory energy E Lab . Several resonances in the compound nuclei15 O,16 O,16 F and 15 F can be observed. The lines are results of the R-matrix calculations using the parameters from Ref.[14] for the15 O and16 O nuclei, from Table I for the16 F nucleus, and from this work for the two resonances observed in15 F.

TABLE II :

 II Experimental proton-neutron interaction energies, Int exp16 N and Int exp

16

F , derived for the

  states in16 N and16 F. Calculated interaction energies Int over 16 F and Int over+C

16

F

Acknowledgments

We thank the GANIL crew for delivering the radioactive beams. We thank B. Bastin, C. Bertulani, P. Descouvemont, M. Dufour, F. Nowacky and P. Van Isacker for fruitful discussions. This work has been supported by the European Community FP6 -Structuring the ERA -Integrated Infrastructure Initiative-contract EURONS n RII3-CT-2004-506065, the IN2P3-IFIN-HH No. 03-33, the IN2P3-ASCR LEA NuAG and partially through FUSTIPEN (French-U.S. Theory Institute for Physics with Exotic Nuclei) under DOE grant number DE-FG02-10ER41700.