1. Loss-Bounded Analysis for Differentiated Services. By Alexander Kesselman and Yishay Mansour Presented By Sharon Lubasz 042824821

2. Abstract Derive tight upper and lower bounds for various settings of the new model. Only trivial bounds could be obtained by traditional competitive analysis. Introduction of a new approach called Loss- Bounded analysis. Network Service offering different levels of Quality of Service (QoS).

3. In the Differentiated Services priority model: packets of different QoS priority have distinct benefit values: For advanced traffic models, there maybe a need for more than two distinct benefits. The lowest benefit of 1. Introduction The highest benefit denoted α, α ≥1. Research of differentiated services For internet traffic products two basics paradigms: The premium service. The assorted service. Basic Paradigms:

4. The premium service model provides the same QoS guarantee as a dedicated line, with a predefined bit rate. The assorted service model traffic flow may exceed its provisioned rate. The excess traffic is not given the same assurance, if any. The premium service model: A premium service traffic is shaped at the entry to the network; hard limited to its provisioned peak rate. The assorted service model: Critical for many application. Most of today’s internet routers work with FIFO buffering policy. The FIFO buffering policy : Simplifies and enable to achieve efficient hardware implementation. Reflects the the nature of the network- the main internet protocol is TCP- optimized in FIFO order. Basic Paradigms (cont.)

5. 1. Stores. A single FIFO queue. Serves online - without knowledge of future packets. Performs two functions: 2. Selectively rejects/preempts packets subject to the buffer. The Model The goal: to maximize the policy’s benefit. V OPT (S) the optimal benefit. L OPT (S) the optimal loss. Definitions: For a sequence of packets S and an online policy A we denote: The subsequence of packets with benefit B, denoted S b. The entire benefit of the sequence denoted V(S) =∑ pєS b(p). The benefit of A on S, denoted V A (S) And the loss of A on S, denoted L A (S). V(S) = V A (S) + L A (S). Denoted the optimal offline policy by OPT.

6. Competitive Analysis The online policy is compared with an optimal offline policy that knows the input in advance. Throughput Competitive and Loss Competitive Throughput Competitive An online policy A, is said to be C-Throughput Competitive if for any input sequence, its benefit constitutes at least a C fraction of the benefit of an optimal offline policy. A is C-throughput Competitive iff for every sequence of packets S, V A (S) ≥ C  V OPT (S). Loss Competitive An online policy A, is said to be C-Loss Competitive if the loss of an optimal offline policy constitute at least a C fraction of its loss. A is C-Loss Competitive iff for every sequence of packets S, L OPT (S) ≥ C  L A (S). A Throughput Competitive guarantee ≠ > Loss Competitive guarantee.

7. Motivation and Loss-Bounded Loss-Competitive guarantee is much more desirable then Throughput Competitive. Unfortunately, only trivial bounds can be obtained by it. Loss-Bounded Analysis, Motivated by this, we propose a new model, called Loss-Bounded Analysis, for estimating loss of an online policy. Loss-Bounded Analysis In C-Loss-Bounded Analysis, the loss of an online policy is upper bounded by the loss of the optimal offline policy plus a C fraction of the benefit of the optimal offline policy. We let this fraction C be the Loss-Bounded ratio of the online policy. Loss-Bounded Analysis provides Throughput Competitive guarantee. One can either maximize the throughput of the policy or minimize its loss. An optimal solution to one problem do not necessarily lead to a good approximation of the other.

8. Loss-Bounded Analysis and Some Intuition The intuition behind Loss-Bounded Analysis is that we try to optimize both parameters simultaneously, by finding an optimal tradeoff between the current gain and the potential loss. A is C-Loss-Bounded iff for every sequence of packets S, L A (S) ≤ L OPT (S) + C  V OPT (S). Definition: The Model - Additions The FIFO buffer can hold B packets. Packets may arrive at any time. Send events are synchronized with time. The system obtains the benefit of the packets it sends. Aiming to maximize the benefit. When a packets arrives, a queuing policy can either reject or except it. Each time unit, a send operation is executed on the first packets in the buffer (first in queue).

9. The Scheduling Policy In that case, a packet with minimal benefit is preempted from the buffer before acceptance of the arriving packet. An arriving packet is accepted if either the buffer is not full or that the buffer is full, and a minimal benefit among the packets in the buffer is less then the benefit of the arriving packet. β-Preemptive Greedy Policy Behaves like a greedy policy, except, that upon acceptance of a packet, β additional packets may be preempted. The preempted packets are the low- benefit packets closest to the transmitting end of the FIFO queue buffer. Overloaded Scheduling Interval 1.Some high benefit packets where rejected during the interval. 2.The longest time interval during which only high benefit packets were sent. Definition:

10. Binary Benefit Values THEOREM THEOREM The greedy policy is 1 /α Loss-Competitive. Proof. Proof. By definition, the cumulative benefit of the lost packets is at most far by a factor of α from the optimal. …for the next Theorem we will need some more tools... LEMMA LEMMA When packets are scheduled according to the √α - preemptive greedy policy, and there are at least B/√ α high benefit packets in the buffer, then at the next time unit, a high benefit packet will be sent. Proof. Proof. There can be B low benefit packets at most (the whole buffer). We know that there are B/√ α high benefit packets, so for each B/√ α packets √ α low benefit packets are preempted… that is the whole B low benefit packets… And so, the only packets left to send are high benefit ones. Auxiliary Lemmas and Claims

11. Auxiliary Lemmas and Claims (cont.) CLAIM CLAIM When packets are scheduled according to the √α - preemptive greedy policy, the number of high benefit packets in the buffer at the time unit preceding the beginning of an overloaded interval [t s,t f ], is at most B/√α. Proof. Proof. There can be two cases: 1. If the queue was empty at the the beginning of the interval, then clearly there where less then B/√α high benefit packets. 2.The interval started after a low benefit packet was sent (time unit t s -1). Lets assume that at time t s -2 there are B/√α or more high benefit packets, then by the lemma a low benefit packet couldn’t have been sent at t s -1. So, it is guaranteed that at t s -2 there are at the most (B/√α)-1 high benefit packets. And so, it is guaranteed that at t s -1 there are at the most (B/√α) high benefit packets.

12. Auxiliary Lemmas and Claims (cont.) and More Theorems More Theorems CLAIM CLAIM When packets are scheduled according to the √ α –Preemptive Greedy Policy the length of an overloaded interval is at least B. Proof. Proof. By definition during the interval at least one high benefit packet must be lost, that could have occurred only when the buffer was full of other high benefit packets. At least those B packets will be send. THEOREM THEOREM The Loss-Bounded ratio of the greedy policy is at most ( α- 1) /α and at least ( α- 1) /2α. Proof. Proof. In the worst case scenario, A benefits at least V OPT (S)/ α, of the optimal gain. And so A losses the maximum possible minus what A gains: L A (S) ≤ L OPT (S) +(1-1/ α) V OPT (S) = L OPT (S) +((α-1)/ α) V OPT (S) V OPT (S) – (V OPT (S)/ α) = (1-1/ α) V OPT (S)

13. More Theorems (cont.) This is the worst case scenario in which A sends a low benefit packet (benefit of 1) for every packet that the offline optimal policy sends a high benefit packet (benefit of α). More intuition for the upper bound: For the lower bound, lets look at the worst case scenario: A burst of B low benefit packets. For B time units every time units a high benefit packet arrives. A burst of B high benefit packets. The Greedy policy: Receives the burst and enters the packets to the buffer. In queues a high benefit packet and transmits the low benefit packets from the head of the buffer. Ignores the burst of the high benefit packets as the buffer is full with high benefit packets. L A (S) = B α …the loss of the last high benefit burst.

14. The optimal policy: Ignores the low benefit burst. Receives and sends the high benefit packets one by one. Receives the high benefit burst. L OPT (S)=B…the loss of the low benefit burst. V OPT (S)=2B α. = > by definition B α ≤ B + C (2B α ). =>=> C = ( α –1)/2( α ) More Theorems (cont.) page 2

15. More Theorems (cont.) page 3 THEOREM THEOREM The Loss-Bounded ratio of the √ α -Preemptive Greedy policy is at most 2/√ α. Proof. Proof. We process the loss of low benefit packet, denoted S 1, and loss of high benefit packets, denotes S α, separately. First we bound the loss of the low benefit packet. Their are two kinds of losses of low benefit packets: 1.A loss of a low benefit packet due to additional preemptions (a high benefit packet arrives while the buffer was full) denote L A extra. 2.A loss of a low benefit packet due to buffer overflow. denote L A ovfl. L A (S 1 ) = L A extra (S 1 ) + L A ovfl (S 1 )

16. First case: Denote S’ α all the high benefit packets that caused the preemption of a low benefit packet. Note that S’ α C S α. – V A (S’ α ) = ( ( L A extra (S 1 ) ) / √α ) α The benefit on all the packets that were received after preempting low benefit packets. The number of high benefit packet that was received after preempting low benefit packets. More Theorems (cont.) page 4 L A extra (S 1 ) ≤ (1/√α )V A (S α ). =>=> L A extra (S 1 ) = (V A (S’ α )√α ) /α= (1/√α ) V A (S’ A ) ≤ (1/√α )V A (S α ) =>=> L A (S 1 ) ≤ (1/√α )V A (S α ) + L OPT (S 1 ). L A ovfl (S 1 ) ≤ L OPT (S 1 ). Second case: The number of low benefit packets that are lost due to the buffer’s capacity constrains - overflow - can be bounded the number of packets that are lost by an optimal offline policy, The constrains are due to the buffer topology, and so the optimal policy will also be affected by it. =>=> L A (S 1 ) = L A extra (S 1 ) + L A ovfl (S 1 ) L A (S 1 ) ≤ (1/√α )V A (S α ) + L A ovfl (S 1 )

17. More Theorems (cont.) page 5 L A (S α ) ≤ (1/√α )V A (S α ) + L OPT (S α ). An optimal offline policy could have send these additional packets. If A were to throw this packets he would have simulated the optimal offline policy. Denoted L A add (S α ). High benefit packets can be lost due to lack of space in the buffer, just like an optimal offline policy, denote this loss L A ncs. L A ncs (S α ) ≤ L OPT (S α ). As shown at the beginning of overloaded interval there are at most B/√ α high benefit packets in the buffer. As shown the length of an overloaded interval is at least B, and so the ratio between the loss and the cumulative benefit of the packets scheduled during an overloaded interval is at most: B√ α / B α = 1/√ α the upper bound on the loss of L A add The lower bound on the benefit of A. ≥ L A add (S α ) / V A (S α ) => L A add (S α ) ≤ (1/√α )V A (S α ). Clearly, L A (S α ) = L A add (S α ) + L A ncs (S α ). L A (S α ) ≤ L A add (S α ) + L OPT (S α ).

18. Summary L A (S α ) ≤ (1/√α )V A (S α ) + L OPT (S α ) L A (S 1 ) ≤ (1/√α )V A (S α ) + L OPT (S 1 ) Lets sum up: + =========================== L A (S) ≤ (2/√α )V A (S ) + L OPT (S) Binary Benefit Values Setting Results Greedy 1/ α(α -1)/ α 2/√ α 0 √ √ α -Prmpt. Greedy As shown the greedy policy achieves 1/ α Loss-Competitive ratio which is the tight upper bound. Thus, the bound for the √α- Preemptive Greedy Policy approaches 0 when α is large.

19. Extended Models All the results of the Loss-Competitive Analysis are trivially extended to the Restricted and Arbitrary benefit models. Restricted Benefit Model !!! As the number of values increases the guarantee is weakened. !!! We extension of the two benefit model to the case of n different values: { α i/n : 0 ≤ i ≤ n} Restricted Benefit Values Setting Results Greedy √ ( α i/n ) -Prmpt.Greedy Impossibility Results 1/ α 0 (α -1)/ (α +1) 1/2√( α 1/n ) (n+2)/√( α 1/n ) + 2/ α 1/n

20. Arbitrary benefit Model We extend the n benefit model to arbitrary benefit model so that for any packet its benefit is between 1 and α. Due to the logarithmic ratio that cannot be better than 1/8logα, which is obviously bigger in a scale than the restricted polynomial bound of 2/√α, arbitrary benefit model is usually inefficient. No online policy under arbitrary benefit values model can have less than logarithmic Loss-Bounded ratio. 1/ α (α -1)/ (α +1) 1/8logα Greedy Impossibility Results Arbitrary Benefit Values Setting Results

21. Concluding Remarks The impossibility results for traditional competitive analysis. Tight lower and upper bounds for FIFO buffer management and packets scheduling. Loss-Bounded Analysis can give much better performances then traditional Competitive Analysis. The model provides simplicity which allows operation at very high speed and without additional equipment. Operator can manage traffic streams in the best way by choosing the appropriate benefit setting. Importance of analysis of the loss of an online policy.