Delay and capacity analysis of structured P2P overlay for lookup service

Abstract

In this paper, we provide an analytical model for the performance study of different structured P2P overlay networks used for lookup service in IP telephony systems. The overlay provides an infrastructure for the lookup service required before an actual voice communication is initiated. Our model captures the performance behavior of such overlays including the mean session set-up delay of a call as well as the system capacity. These parameters reflect how good an IP telephony overlay is performing. We formulate the system as a queuing network. We idealize Chord routing semantics to extract useful observations to obtain closed form expressions for the session setup delay and capacity as a function of the number of participating supernodes (SN). The analysis also answers the question of finding an optimum number of SN for minimum session setup delay.

Appendices

Appendix 1: proof of Proposition 5

Proposition

In the load-balanced Chord Model with $N\!=\!2^c$ and $K=2^k$, each peer will have $c$ neighbors.

Proof

To prove this result, we assert the following three statements with proofs:

Statement 1: Any SN with SN id $x+2^{i-1} T$ for $1\leqslant i\leqslant c$ is a finger entry and a distinct neighbor of a given SN.

Proof of the statement 1: We represent $T=\frac{2^k}{2^c}=2^{k-c}$, the partition of keyspace belonging to each SN. For a SN $X$ with node id $x$, all other SNs are occupying the ids: $(x+T)\%K,(x+2T)\%K,(x+3T)\%K\cdots ,(x+iT)\%K,\cdots$. $\%$ represents the modulo operation. But not all of them fall in the neighbor list of SN $X$.

We see that $x+2^{i-1} T = x+ 2^{k-c+i}$ is an element of the set $X=\{x+2^{i-1} : i= 1,2,3,\cdots , k\}$ and this $S$ is the set of all fingers anticipated by the Chord routing. (To learn about what are the possible finger entries in Chord, refer to [25]). The maximum entry of the finger list is when $2^{k-c-i} = 2^{k-1}$. i.e. $i=c-1$. This proves the statement 1.

Statement 2: No other SNs fall in the set $X=\{x+2^{i-1} : i= 1,2,3,\cdots , k\}$.

Proof of the statement 2: First we assume that this statement is false. For the above statement to be false, we require both of the following equations to be satisfied.

$$\begin{aligned} x+nT&=x+2^{i-1}\end{aligned}$$

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$$\begin{aligned} n&\ne 2^l \text { For any } l \in \{1,2,3,4, \cdots c-1\} \end{aligned}$$

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But, it is impossible to satisfy both equations together for $T=2^{k-c}$. Thus statement 2 is valid.

Statement 3: No reference is ever made by the finger pointer to a key that lies between neighbor SNs with node id $x+2^{j-1} T$ and $x+2^{j} T$ for $1\leqslant j\leqslant c$.

Proof of the statement 3: We determine the entry $i_1$ in the finger table corresponding to node ID $x+2^{j-1} T$ from the equation below

$$\begin{aligned} x+2^{j-1} T&=x+2^{i_1-1}\\ i_1&={j}+{k-c}=j+k-c \end{aligned}$$

Also the corresponding entry $i_2$ in the finger table corresponding to node ID $x+2^{j} T$ is solved by using the equation below

$$\begin{aligned} x+2^{j-1} T&=x+2^{i_2-1}\nonumber \\ i_2&=j+k-c+1 \end{aligned}$$

The next entry after $i_1$ in finger entry table is $i_1+1$ pointing to the key $x+2^{i_1+1-1}$. This is the entry corresponding to $i_2$. It means that the next finger pointer after $i_1$ is $i_2$ itself. There are no referrals to any key in between.

Statement 1 means that $c$ SNs having id $x+2^i T$ for $0 \leqslant i \leqslant c-1$ fall in the neighbor list of SN $X$. Statement 2 says that no other SNs fall as the neighbor of SN $X$. Statement 3 shows that there would be no pointer references to a key between the two fingers as determined in statement 1. The three statements then together mean that there are exactly $c$ neighbors of SN $X$. $\square$

Appendix 2: proof of Proposition 6

Proposition: Prove that in a load-balanced Chord, a lookup being forwarded to the $i$th neighbor of SN $X$ is forwarded to one of the first $i-1$ neighbors only.

Proof

In order to prove this result, we first assert the following statement.

Statement 4: The $i$th neighbor of the $i$th neighbor of node $X$ is the $(i+1)$th neighbor of node $X$.

Proof of the statement 4: We represent the node id of the $i$th neighbor of node $X$ with node id $x$ as $N_i(x)$. Then from previous argument in statement 1 of Appendix 1, we have

$$\begin{aligned} N_i(x)=x+2^{i-1} T \end{aligned}$$

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The $i$th neighbor of the $i$th neighbor of node $X$ is represented as $N_i (N_i(x))$. We obtain the following

$$\begin{aligned} N_i (N_i(x))&=N_i(x+2^{i-1} T)=x+ 2^ {i-1} T + 2^ {i-1} T \nonumber \\&=x + T (2^{i-1}+2^{i-1})= x+2^i T \end{aligned}$$

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But, from 38, we have

$$\begin{aligned} N_{i+1}(x) = x + 2^ {i+1-1} \end{aligned}$$

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From 39 and 40, we conclude that statement 4 is true.

Using this statement 4, for any lookup forwarded by node $X$ to its $i$th neighbor $Y$, we can say that $Y$ will never forward the lookup to its $i$th neighbor $Z$. This is because if it had to, $X$ would have directly forwarded to its $i+1$th neighbor (node $Z$) following the greedy algorithm. The greedy behavior of routing also prohibits any subsequent forward by node $Y$ beyond its $i$th neighbor because if any such nodes existed, $Z$ would be chosen by $X$ at first place. $\square$

Appendix 3: proof of Proposition 7

Proposition

Prove that each SN in the load-balanced Chord has $2^{c-b}$ E$b$-NRSN.

Proof

In the Lookup Routing Path Tree (LRPT) of any SN $i$, a SN in the $j$th branch of LRPT can have one more SN if and only if $j \ge b$.

We define $\Lambda _b(i)$ as the number of E$b$-NRSN in the $i$th branch of the LRPT of the SN under consideration. Then, we have the following:

$$\begin{aligned} \Lambda _b(i)&=0 \text { For } 0 \leqslant i < b \nonumber \\ \Lambda _b(b)&=1 \end{aligned}$$

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The LRPT has a general recurrence as described earlier. The recursion leads to the fact that the number of E$b$-NRSN in the $i+1$th branch is the sum of the numbers in the previous branches. i.e.

$$\begin{aligned} \Lambda _b(i+1)&= \sum \limits _{j=0}^{i} {\Lambda _b(j)}=\sum \limits _{j=b}^{i} {\Lambda _b(j)}\nonumber \\&=\Lambda _b(b)+\sum \limits _{j=b+1}^{i} {\Lambda _b(j)} \text { For all }i > b \end{aligned}$$

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This recurrence and the initial condition of $\Lambda _b(b)=1$ means that the expression for $\Lambda _b(i+1)$ is expressed as:

$$\begin{aligned} \Lambda _b(i+1)&=1+1+2+2^2+2^3+\cdots +2^{i-b}=1\nonumber \\&+\frac{2^{i-b}-1}{2-1}\nonumber \\&=2^{i-b} \end{aligned}$$

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Then we have

$$\begin{aligned} \Lambda _b(c)=1+\sum \limits _{i=b+1}^{c-1}\Lambda _b(i)=2^{c-b-1} \end{aligned}$$

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Now, the total number of E$b$-NRSN (i.e. $n_b$) is the sum of all these. i.e

$$\begin{aligned} n_b&= \sum \limits _{i=b}^{c}\Lambda _b(i)=1+\sum \limits _{i=b+1}^{c-1} \Lambda _b(i)+\Lambda _b(c)\nonumber \\&=2\Lambda _b(c)=2\times 2^{c-b-1}=2^{c-b} \end{aligned}$$

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$\square$

Appendix 4: derivation of lookup hop distribution of load-balanced chord DHT

Let, random variable $S$ denote the number of SNs visited by a lookup originating at one of the SNs.

we define,

$$\begin{aligned} P\{ S = i\} \!=\! {\text {Prob}}\{ \text {A Lookup visits }i{\text { SNs before absorption}}\} \end{aligned}$$

By observing the lookup routing path tree in Fig. 5, and defining $X(i)$ as the number of nodes in the tree in Fig. 5 that are $(i-1)$ hop away from the originating SN, we note the followings:

$$\begin{aligned} X(1)&=1 \quad X(2)=c \quad X(3)=\sum \limits _{i = 1}^{c - 1} i \quad X(4)=\sum \limits _{i = 1}^{c - 2} {\sum \limits _{j = 1}^i j } \nonumber \\&\cdots \nonumber \\ X(\kappa )&=\underbrace{\sum \limits _{a = 1}^{c - (\kappa - 2)} {\sum \limits _{b = 1}^a { \cdots \sum \limits _{i = 1}^h {\sum \limits _{j = 1}^i {} j} } } }_{(\kappa - 2){\text { sums of sums}}} \quad for 1 \le \kappa \le (c+1) \end{aligned}$$

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For any lookup originating at one of the SNs, the probability of a lookup being absorbed after visiting $i$ SNs is proportional to $X(i)$ as all the nodes in the root are equiprobabally absorbing a lookup.We get,

$$\begin{aligned} P\{ S = i\}=\frac{X(i)}{N} \end{aligned}$$

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Now, the expected number of SNs visited before a lookup is absorbed is:

$$\begin{aligned} E\{S\}= \overline{S}&=\sum \limits _{i = 1}^{c + 1}i P\{ S = i\}\nonumber \\&=1+\frac{c}{2} \end{aligned}$$

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