Constrained max-min fair scheduling of variable-length packet-flows to multiple servers

Abstract

In this paper, we study a multi-server queuing system wherein each user is constrained to get service only from a specified subset of servers. Fair packet scheduling in such a setting poses novel challenges that we address in this paper. Specifically, we observe that max-min fair allocation of the available resource over different servers (notably bandwidth) in the presence of placement constraints results in different levels of fair service-rates. To achieve the max-min fair service rates, we propose a novel packet scheduler which is inspired by the deficit-round robin (DRR) algorithm. The scheduler allocates tokens to flows in a round-by-round manner, where token allocation to flows at the beginning of each round is weighted max-min fair. So, we have called it multi-server max-min fair DRR (MSMF-DRR). The performance of the MSMF-DRR algorithm in terms of achieving fairness is shown through a worst-case performance analysis. In addition to analytical results, numerical experiments are also carried out to illustrate service isolation and the delay guarantee that are provided by the algorithm. Generally, a scheduler for such a constrained multi-server queuing system can be applicable in many modern data-networking applications, especially in cloud computing wherein virtual machines and/or processes vie for different IT resources distributed over heterogenous servers, while different processes may have preferences over servers owing to their quality-of-service requirements and the heterogeneity of servers.

Appendix

Proof of Theorem 2 - Proof of only if.

Let {α _{n, i} } denote an optimal solution to problem $\mathcal {P}_{1}$.The feasibility constraint in Eq. 13 requires that α _{n, i} = 0 when δ _{n, i} = 0. Let $\mathcal {K}_{1}$ denote a subset of active servers such that:

$$ \mathcal{K}_1:=\{i\in\mathcal{K}|\delta_{n,i}=1,\text{ for some }n\in\mathcal{B}\}. $$

(39)

Without loss of generality we can assume that α _{n, i} = 0 for $n\notin \mathcal {B}$or$i\notin \mathcal {K}_{1}$. In fact, we canformulate $\mathcal {P}_{1}$only in terms of {α _{n, i} } such that $i\in \mathcal {K}_{1}$, $n\in \mathcal {B}$, and δ _{n, i} = 1. The Lagrangian function for this problem can be written as:

$$\begin{array}{@{}rcl@{}} L(\alpha,\lambda,\nu)& = &\sum\limits_{n\in\mathcal{B}}\phi_{n} g(\tilde{r}_{n})\\ &&\!+ \sum\limits_{i\in\mathcal{K}_{1}}\lambda_{i}\left( 1\!-\sum\limits_{n\in\mathcal{B}}\alpha_{n,i}\right)\!\sum\limits_{n\in\mathcal{B},i\in\mathcal{K}_{1}}\nu_{n,i}\alpha_{n,i}, \end{array} $$

where λ _i ≥ 0and ν _{n, i} ≥ 0 denote the Lagrange multipliers associated with inequality constraints in Eqs. 11 and 12, respectively, and $\tilde {r}_{n}:={\sum }_{i}\alpha _{n,i}c_{i}/\phi _{n}$. For problem $\mathcal {P}_{1}$, with aconcave objective function and a convex feasibility region, KKT conditions describe necessary and sufficient conditions for optimality [4].Specifically, {α _{n, i} } is an optimal solution to this problem iff there exists a set of Lagrange multipliers {λ _i }and {ν _{n, i} }, for everyserver $i\in \mathcal {K}_{1}$ and each flow $n\in \mathcal {B}$for which δ _{n, i} = 1, such that:

$$\begin{array}{@{}rcl@{}} && \frac{\partial L(\alpha,\lambda,\nu)}{\partial\alpha_{n,i}}=c_{i}g^{\prime}({\tilde{r}_{n}})-\lambda_{i}+\nu_{n,i}=0, \end{array} $$

(40)

$$\begin{array}{@{}rcl@{}} && \lambda_{i}\left( {\sum}_{n}\alpha_{n,i}-1\right)=0, \end{array} $$

(41)

$$\begin{array}{@{}rcl@{}} && \nu_{n,i}\alpha_{n,i}=0. \end{array} $$

(42)

Given that g(⋅) is a strictly concave and increasing function, it follows that g′ (⋅) > 0. Hence, Eq. 40 implies that λ _i > ν _{n, i} , for each flow $n\in \mathcal {B}:\delta _{n,i}=1,~\forall i\in \mathcal {K}_{1}$. As a result, it follows that $\lambda _{i}>0, \forall i\in \mathcal {K}_{1}$. Accordingly, Eq. 41 implies that the inequality constraints in Eq. 11 are active, i.e.,

$$ \sum\limits_n\alpha_{n,i}=1, \forall i\in\mathcal{K}_1. $$

(43)

Since g(⋅) is a strictly concave function, g′(⋅) is strictly decreasing, and as a result it is invertible. The inverse of g′(⋅) is also a strictly decreasing function which we denote it by h(⋅). Accordingly, Eq. 40 can be rewritten as:

$$ \tilde{r}_n=h\left( \frac{\lambda_i-\nu_{n,i}}{c_i}\right), ~\forall n\in\mathcal{B}: \delta_{n,i}=1. $$

(44)

Now consider two active flows n and m for which δ _{n, i} = δ _{m, i} = 1 and α _{n, i} > 0. Since α _{n, i} > 0, then Eq. 42 implies that ν _{n, i} = 0. Therefore:

$$ \tilde{r}_n=h\left( \frac{\lambda_i}{c_i}\right)\le h\left( \frac{\lambda_i-\nu_{m,i}}{c_i}\right)=\tilde{r}_m $$

(45)

where the above inequality follows from the fact that h(⋅) is a decreasing function. Given that Eqs. 43 and 45 are established, Theorem 1 implies that {α _{n, i} } is a set of max-min fair allocation parameters. □

Proof of if

Consider a set of max-min fair allocation parameters, {α _{n, i} }. The fact that {α _{n, i} } satisfies max-min fairness implies that α _{n, i} = 0 if δ _{n, i} = 0 (Otherwise, we may increase $\tilde {r}_{m}$for some flow m with δ _{m, i} = 1, without decreasing $\tilde {r}_{n}$. This contradicts the fact that {α _{n, i} } satisfies max-min fairness.). Also, we can assume that α _{n, i} = 0 if $n\notin \mathcal {B}$ or $i\notin \mathcal {K}_{1}$(c.f. Eq. 39). We will show that KKT conditions Eqs. 40–42 are satisfied for every server $i\in \mathcal {K}_{1}$ and each flow $n\in \mathcal {B}$for which δ _{n, i} = 1.

The fact that {α _{n, i} } satisfies max-min fairness implies that ${\sum }_{n}\alpha _{n,i}=1,~\forall i\in \mathcal {K}_{1}$(Otherwise, we may increase $\tilde {r}_{m}$for some flow m with δ _{m, i} = 1, without decreasing the allocated service rates to other flows.). Hence, Eq. 41 is established for all servers $i\in \mathcal {K}_{1}$. For each server $i\in \mathcal {K}_{1}$, there exists some flow n for which α _{n, i} > 0. According to Theorem 1, it follows that $\tilde {r}_{n}=\tilde {r}_{p},~\forall p:\alpha _{p,i}>0$. Hence, we can choose $\lambda _{i}:=c_{i}g^{\prime }(\tilde {r}_{n})$. If we consider an arbitrary flow for which δ _{m, i} = 1, Theorem 1 implies that $\tilde {r}_{m}\ge \tilde {r}_{n}$. Therefore, we can choose $\nu _{m,i}=\lambda _{i}-c_{i}g^{\prime }(\tilde {r}_{m})=c_{i}[g^{\prime }(\tilde {r}_{n})-g^{\prime }(\tilde {r}_{m})]\ge 0$, which results in ν _{m, i} = 0 when α _{m, i} > 0. Hence, Eqs. 40 and 42 are established for all active flows for which δ _{m, i} = 1. □

Proof of Lemma 1

Let ${\mathcal Z}_{i}:=\{n\in \mathcal {B}\mid \alpha _{n,i}^{(l-1)}=0\}$. The Lagrangian function for ${\mathcal P}_{2}$ is:

$$ L({\alpha},{\lambda},{\nu}):=\frac{1}{2}\|\textbf{d}^{\prime}_i-\textbf{d}_i\|^2+\lambda\sum\limits_{n\in\mathcal{B}}d^{\prime}_{n,i}-\sum\limits_{n\in{\mathcal Z}_i}\nu_{n,i}d^{\prime}_{n,i}. $$

(46)

For the convex optimization problem $\mathcal {P}_{2}$, KKT conditions describe necessary and sufficient conditions for optimality. Specifically,

$$\begin{array}{@{}rcl@{}} &&d^{\prime}_{n,i}-d_{n,i}+\lambda-\nu_{n,i}\textbf{1}_{n\in{\mathcal Z}_{i}}=0,~\forall n\in\mathcal{B},~~ \end{array} $$

(47)

$$\begin{array}{@{}rcl@{}} &&d^{\prime}_{n,i}\nu_{n,i}=0,~\forall n\in{\mathcal Z}_{i}, \end{array} $$

(48)

$$\begin{array}{@{}rcl@{}} &&\sum\limits_{n\in\mathcal{B}}d^{\prime}_{n,i}=0. \end{array} $$

(49)

Finding $d^{\prime }_{n,i}$from Eq. 47 and substituting it into Eq. 49 results in:

$$ \lambda=\frac{{\sum}_{n\in\mathcal{B}}d_{n,i}+{\sum}_{n\in{\mathcal Z}_i}\nu_{n,i}}{|\mathcal{B}|}, $$

(50)

where $|\mathcal {B}|={\sum }_{n\in \mathcal {B}}1$. Accordingto Eq. 48, if $d^{\prime }_{n,i}>0$then ν _{n, i} = 0. Therefore, Eq. 48 implies that:

$$ d^{\prime}_{n,i} = \left[d_{n,i}-\frac{{\sum}_{m\in\mathcal{B}}d_{m,i}+{\sum}_{m\in{\mathcal Z}_i}\nu_{m,i}}{|\mathcal{B}|}\right]^+. $$

(51)

Let $\bar {\mathcal {Z}}_{i}:=\{m\in {\mathcal Z}_{i}\mid d^{\prime }_{m,i}=0\}$. Accordingto Eq. 48, if ν _{n, i} > 0 then $n\in \bar {\mathcal {Z}}_{i}$. Therefore, ${\sum }_{n\in {\mathcal {Z}}_{i}}\nu _{n,i} = {\sum }_{n\in \bar {\mathcal {Z}}_{i}}\nu _{n,i}$. After summing bothsides of Eq. 47 over $n\in \bar {\mathcal {Z}}_{i}$and doing some manipulations, it follows that:

$$ \sum\limits_{n\in{\mathcal{Z}}_i}\nu_{n,i}=\frac{|\bar{\mathcal{Z}}_i|{\sum}_{n\in\mathcal{B}}d_{n,i}-|\mathcal{B}|{\sum}_{n\in\bar{\mathcal{Z}}_i}d_{n,i}}{|\mathcal{B}|-|\bar{\mathcal{Z}}_i|}, $$

(52)

where $|\bar {\mathcal {Z}}_{i}|={\sum }_{n\in \bar {\mathcal {Z}}_{i}}1$. Substituting ${\sum }_{n\in {\mathcal {Z}}_{i}}\nu _{n,i}$from Eq. 52 into Eq. 51 results in:

$$ d^{\prime}_{n,i}=\left[d_{n,i}-\frac{{\sum}_{m\in {\mathcal A}_i}d_{m,i}}{{\sum}_{m\in {\mathcal A}_i}1}\right]^+. $$

(53)

where $\mathcal {A}_{i}:=\{m\in \mathcal {B}\mid \alpha _{m,i}^{l-1}>0\text { or }d^{\prime }_{m,i}>0\}$. □

A sketch of the Proof of Lemma 2

According to the Constrained Optimality Theorem, $\mathbf {d}^{\prime \ast }_{i}$is the optimal solution to $\mathcal {P}_{2}$if and only if [4]:

$$ (\mathbf{d}^{\prime *}_i-\mathbf{d}_i)^T(\mathbf{d}^{\prime}_i-\mathbf{d}^{\prime*}_i)\ge 0, $$

(54)

for all feasible directions $\mathbf {d}^{\prime }_{i}$satisfying theconstraints in Eqs. 15–17. By substituting $\mathbf {d}^{\prime }_{i}=0$into Eq. 54, it follows that $\mathbf {d}_{i}^{T}\mathbf {d}^{\prime *}_{i}\ge \|d^{\prime *}_{i}\|^{2}$, which shows that $\mathbf {d}^{\prime *}_{i}$is an ascent direction. According to backtracking line search criteria in Eq. 20, $G(\alpha ^{(l)})-G(\alpha ^{(l-1)})\ge \frac {1}{2}\beta {\sum }_{i}\mathbf {d}_{i}^{T}\mathbf {d}^{\prime *}_{i}$. Hence, the basic convergence theorem for backtracking line search implies that the sequence {α ^(l)} linearly converges tosome feasible point α ^∗, where $\forall i,~\|\mathbf {d}^{\prime *}_{i}\|\rightarrow 0$.

Let $\mathbf {d}^{\prime }=[\mathbf {d}^{\prime }_{i}]$denote an arbitraryfeasible direction at α ^∗. So, we can find β > 0 such that α ^∗ + β d′ is a feasiblepoint. Given that G(⋅) is a concave function, it follows that:

$$\begin{array}{@{}rcl@{}} G(\alpha^{*}+\beta \mathbf{d}^{\prime}) &\le& G(\alpha^{*})+\beta \mathbf{d}^{T}\mathbf{d}^{\prime} \end{array} $$

(55)

where d = [d _i ] is the gradient vector at α ^∗. Given that $\mathbf {d}^{\prime }_{i}$ and $\mathbf {d}^{\prime *}_{i}$both are feasible directions satisfying Eqs. 15–17, it follows that $\mathbf {d}^{\prime }_{i}+\mathbf {d}^{\prime *}_{i}$also satisfies the constraints in Eqs. 15–17and therefore is a feasible direction. If we substitute $\mathbf {d}^{\prime }_{i}+\mathbf {d}^{\prime *}_{i}$ instead of $\mathbf {d}^{\prime }_{i}$ in Eq. 54, itfollows that $(\mathbf {d}_{i}-\mathbf {d}^{\prime *}_{i})^{T}\mathbf {d}^{\prime }_{i}\le 0$. Applying this into Eq. 55 results in:

$$\begin{array}{@{}rcl@{}} G(\alpha^{*}+\beta \mathbf{d}^{\prime}) &\le& G(\alpha^{*})+\beta \mathbf{d}^{\prime T}\mathbf{d}^{\prime*}\\ &\doteq & G(\alpha^{*}) \end{array} $$

(56)

where the equality follows from the fact that ∀i, ∥d ^′∗ _i ∥→ 0.The inequality in Eq. 56 implies that α ^∗is a local optimum point for $\mathcal {P}_{1}$. Thefact that α ^∗ is globally optimal followsfrom convexity of the problem. □

Proof of Theorem 3

Let ζ _i,1 denote the last instant of time until t ₁, ζ _i,1 ≤ t ₁, at which a cycle expires at server i. Let ζ _i,0 denote the beginning of this cycle. The remaining released tokens for every flow n from server i at ζ _i,1, ρ _{n, i} (ζ _i,1) is less than L.⁷ Let ρ _{n, i} (ζ _i,1, t ₂)denote the amount of tokens released for flow n from server i during [ζ _i,1, t ₂). The amount of service received by flow n during [t ₁, t ₂), W _{n, i} (t ₁, t ₂) is less than or equal to:

$$\begin{array}{@{}rcl@{}}W_{n,i}(t_{1},t_{2}) & \le & \rho_{n,i}(\zeta_{i,1},t_{2}) + \rho_{n,i}(\zeta_{i,1})-\rho_{n,i}(t_{2})\\ & \le & \rho_{n,i}(\zeta_{i,1},t_{2}) + L , \end{array} $$

(57)

where the second inequality follows from the fact that ρ _{n, i} (t ₂) ≥ 0 and ρ _{n, i} (ζ _i,1) < L. On the other hand, ρ _{n, i} (ζ _i,1, t ₂) is upper bounded by the amount oftokens that are allocated during [ζ _i,1, t ₂) plus the tokens that are currently allocated at ζ _i,1, i.e.,

$$ \rho_{n,i}(\zeta_{i,1},t_2)\le \theta_{n,i}(\zeta_{i,1})+\Theta_{n,i}(\zeta_{i,1},t_2) $$

(58)

Any tokens that are allocated from server i until ζ _i,0, are released at ζ _i,0. Therefore, 𝜃 _{n, i} (ζ _i,1) is tokens that are allocated during (ζ _i,0, ζ _i,1). Therefore:

$$ \theta_{n,i}(\zeta_{i,1})+\Theta_{n,i}(\zeta_{i,1},t_2)=\Theta_{n,i}(\zeta_{i,0}^+,t_2) $$

(59)

Assume that token allocations are updated l times during (ζ _i,0, t ₂). Specifically, let τ ₁, τ ₂, ... , τ _l ∈ (ζ _i,0, t ₂) denote the instants of updating. Hence:

$$ \Theta_{n,i}(\zeta_{i,0}^+,t_2) = \sum\limits_{h=1}^l\Theta_{n,i}(\tau_h,\tau_h^+) $$

(60)

where $\Theta _{n,i}(\tau _{h},\tau _{h}^{+})$ is the amount of tokens allocated to each flow n from server i in the h ^th token allocation, 1 ≤ h ≤ l:

$$ \Theta_{n,i}(\tau_h,\tau_h^+)=r_{n,i}^{(h)}(\tau_h-\tau_{h-1}) $$

(61)

where τ ₀is the lastinstant of time until ζ _i,0, τ ₀ ≤ ζ _i,0, at which token allocationsare updated, and $r_{n,i}^{(h)}=\alpha _{n,i}^{(h)}c_{i}$is the service rate that is assigned to flow n from server i in the h ^thallocation. The allocated tokens at τ ₀are released at the beginning of thecycle starting at ζ _i,0. This cycle continuespast ζ _i,1, where a new cycle starts. Hence, τ ₀should be greater than or equal to ζ _i,1 − T ₀.⁸ With the same line of arguments, it follows that ζ _i,1 ≥ t ₁ − T ₀.As a result, τ ₀ ≥ t ₁ − 2T ₀. Since the set of activeflows does not change during ((t ₁ − 2T ₀)⁺, t ₂),the allocation parameters, and hence the allocated service rates, are the same for τ _h > τ ₀, that is$r_{n,i}^{(h)}=r_{n,i}$for 1 ≤ h ≤ l. If we sum Eq. 61 over different h, 1 ≤ h ≤ l,it follows that:

$$\begin{array}{@{}rcl@{}} \Theta_{n,i}(\zeta_{i,0}^{+},t_{2}) & = & \sum\limits_{h=1}^{l}\Theta_{n,i}(\tau_{h},\tau_{h}^{+})\\ & = & r_{n,i}(\tau_{l}-\tau_{0})\\ &\le& r_{n,i}(t_{2}-t_{1}+2T_{0}) \end{array} $$

(62)

where the last inequality follows from the fact that τ _l ≤ t ₂and τ ₀ ≥ t ₁ − 2T ₀. From Eqs. 57–59, and 62, it follows that:

$$\begin{array}{@{}rcl@{}} W_{n}(t_{1},t_{2}) & = & \sum\limits_{i:\delta_{n,i}=1}W_{n,i}(t_{1},t_{2})\\ &\le& \sum\limits_{i:\delta_{n,i}=1}[r_{n,i}(t_{2}-t_{1}+2T_{0})+L]\\ & = & r_{n}(t_{2}-t_{1})+\left[2r_{n}T_{0}+L\sum\limits_{i}\delta_{n,i}\right] \end{array} $$

(63)

To complete the proof, we also need to prove that:

$$ W_n(t_1,t_2)\ge r_n(t_2-t_1)-\left( 2r_nT_0+L{\sum}_i\delta_{n,i}\right) $$

(64)

The lower bound in Eq. 64 can be shown in the same way. Alternatively, the reader may follow the more general proof of Theorem4. □

Proof of Lemma 3

Let $\hat {t}_{2}\in [t_{1},t_{2}]$denote the last instant of time until t ₂, $\hat {t}_{2}\le t_{2}$, at which ${x}_{i}(\hat {t}_{2})\neq 0$for some server i. If there is not such instant of time let $\hat {t}_{2}=t_{1}$. In case that $\hat {t}_{2}<t_{2}$, x _i (t) = 0for every $t\in (\hat {t}_{2},t_{2}]$. Let us assume that token allocations are updated during $[t_{1},\hat {t}_{2})$for l times. Specifically, let τ ₁, τ ₂, ... , τ _l denote the instants of updates during $[t_{1},\hat {t}_{2})$. Let τ ₀, τ ₀ < t ₁, denote the last token allocation before t ₁. The total number of tokens allocated to flow n during $[t_{1},\hat {t}_{2})$, $\Theta _{n}(t_{1},\hat {t}_{2})$is:

$$\begin{array}{@{}rcl@{}} \Theta_{n}(t_1,\hat{t}_2) & = & \sum\limits_{h=1}^l\Theta_n(\tau_h,\tau_h^+)\\ & = & \sum\limits_{h=1}^l\sum\limits_i\alpha_{n,i}^{(h)}s_i(\tau_h), \end{array} $$

(65)

where $\Theta _{n}(\tau _{h},\tau _{h}^{+})$is the amount of tokens allocated to flow n in the h ^th update, and $\{\alpha _{n,i}^{(h)}\}$are the corresponding allocation parameters. We define:

$$\begin{array}{@{}rcl@{}} r_n{(\tau_h)}: & = &\sum\limits_i\alpha_{n,i}^{(h)}s_i(\tau_h)/(\tau_h-\tau_{h-1})\\ & = & \sum\limits_i\alpha_{n,i}^{(h)}\bar{c}_i(\tau_h), \end{array} $$

(66)

as the assigned service rate to flow n in the hth update, where $\bar {c}_{i}(\tau _{h})$- as defined by Eq. 27- is the average capacity of server i observed over interval (τ _h−1, τ _h ). According to Lemma 4 (c.f. Eq. 70):

$$\begin{array}{@{}rcl@{}} \sum\limits_i\alpha_{n,i}^{(h)}s_i(\tau_h) & = & r_n{(\tau_h)}[\tau_h-\tau_{h-1}]\\ &\ge& {\int}_{\tau_{h-1}+\bar{T}}^{\tau_h+\bar{T}}\hat{r}_n(z)dz, \end{array} $$

(67)

where $\hat {r}_{n}(z)$is the assigned service rate to flow n when a set of servers with the capacities {x _i (z)}- as defined in Eq. 31-are allocated to the set $\mathcal {B}$of active flows in a max-min fair manner. The inequality in Eq. 67 along with Eq. 65 imply that:

$$\begin{array}{@{}rcl@{}} \Theta_{n}(t_1,t_2) &\ge& \sum\limits_{h=1}^l{\int}_{\tau_{h-1}+\bar{T}}^{\tau_h+\bar{T}}\hat{r}_n(z)dz\\ & = & {\int}_{\tau_{0}+\bar{T}}^{\tau_l+\bar{T}}\hat{r}_n(z)dz\\ &\ge& {\int}_{t_{1}+\bar{T}}^{\tau_l+\bar{T}}\hat{r}_n(z)dz, \end{array} $$

(68)

where the last inequality follows from the fact that τ ₀ < t ₁. The fact that ${x}_{i}(\hat {t}_{2})\neq 0$for someserver i, implies that c _i (τ) > 0 for $\tau \in (\hat {t}_{2}-2\bar {T},\hat {t}_{2}]$, i.e., server i is continuously active during $(\hat {t}_{2}-2\bar {T},\hat {t}_{2}]$. Hence, the amount ofservice given by server i during $[\hat {t}_{2}-\bar {T},\hat {t}_{2})$is greater than or equal to $\sigma _{i}=c_{i}^{\min }\bar {T}$(c.f. Eq. 30). On the other hand, the total number of tokens allocated from server i at $\hat {t}_{2}-\bar {T}$(whether released or notreleased) is upper bounded by σ _i . Therefore, token allocations should be updated at least one time during $[\hat {t}_{2}-\bar {T},\hat {t}_{2})$, that is $\tau _{l}\geq \hat {t}_{2}-\bar {T}$. This along with Eq. 68 result in:

$$\begin{array}{@{}rcl@{}} \Theta_{n}(t_1,t_2) &\ge& {\int}_{t_{1}+\bar{T}}^{\hat{t}_2}\hat{r}_n(z)dz,\\ & = & {\int}_{t_1+\bar{T}}^{t_2}\hat{r}_n(z)dz \end{array} $$

(69)

where the last equality follows from the fact that x _i (t) = 0, $~\forall i,~\forall t\in (\hat {t}_{2},t_{2}]$, which impliesthat $\hat {r}_{n}(t)=0,~\forall t\in (\hat {t}_{2},t_{2}]$. □

Lemma 4

Let τ _h denote an instant of time where token allocations are updated. Let τ _h−1 denote the last token allocation before τ _h . The assigned service rate to each flow n at τ _h - as defined by Eq. 66 - is greater than or equal to:

$$ r_n{(\tau_h)}\ge {\int}_{\tau_{h-1}+\bar{T}}^{\tau_h+\bar{T}}\hat{r}_n(z)dz/(\tau_h-\tau_{h-1}), $$

(70)

Proof

Let define $\mathcal {L}_{i}(\tau _{h-1},\tau _{h})$aslength of the interval (τ _h−1, τ _h ) over which server i is active. Since token allocations are not updated during (τ _h−1, τ _h ), it follows that: $\mathcal {L}_{i}(\tau _{h-1},\tau _{h})\le \bar {T},~\forall i$. Weconsider two different cases:

• Case 1: $\tau _{h}-\tau _{h-1}>\bar {T}$. In this case, for each server i and for every z ₀ ∈ (τ _h−1, τ _h ), there exists some y ₀ ∈ (τ _h−1, τ _h ) such that: c _i (y ₀) = 0 and $|z_{0}-y_{0}|\le \bar {T}$(otherwise $\mathcal {L}_{i}(\tau _{h-1},\tau _{h})>\bar {T}$). Hence, for every z ₀ ∈ (τ _h−1, τ _h ):

  $$\begin{array}{@{}rcl@{}} x_i(z_0+\bar{T}) & = & \inf\{c_i(\tau)\mid\tau\in(z_0-\bar{T},z_0+\bar{T}]\}\\ & = & c_i(y_0)=0. \end{array} $$

  Accordingly, x _i (z) = 0, ∀i, and $\forall z\in (\tau _{h-1}+\bar {T},\tau _{h}+\bar {T})$. In this case, the lower boundin Eq. 70 holds trivially as $r_{n}(\tau _{h})\ge \hat {r}_{n}(z)=0$ for every $z\in (\tau _{h-1}+\bar {T},\tau _{h}+\bar {T})$.

• Case 2: $\tau _{h}-\tau _{h-1}\le \bar {T}$. In this case, for every $z\in (\tau _{h-1}+\bar {T},\tau _{h}+\bar {T})$:

  $$\begin{array}{@{}rcl@{}} x_i(z) & = & \inf\{c_i(\tau)\mid\tau\in(z-2\bar{T},z]\}\\ &\le& \inf\{c_i(\tau)\mid\tau\in(\tau_{h-1},\tau_h)\} \le \bar{c}_i(\tau_h),~\forall i. \end{array} $$

  This implies that:⁹ $\hat {r}_{n}(z)\le r_{n}{(\tau _{h})},~\forall z\in (\tau _{h-1}+\bar {T},\tau _{h}+\bar {T})$. Integrating on both sides of this inequality over the interval $(\tau _{h-1}+\bar {T},\tau _{h}+\bar {T})$ and dividing both sides by (τ _h − τ _h−1) result in Eq. 70.

□

Proof of Theorem 4

Let σ _i,1 denote the last instant of time until t ₂, σ _i,1 ≤ t ₂, at which a cycle expires at server i. Let σ _i,0 denote the beginning of this cycle. The remaining released tokens for every flow n from server i at σ _i,1, ρ _{n, i} (σ _i,1) is less than L. Let ρ _{n, i} (t ₁, σ _i,1) denote the amount of tokens released for flow n from server i during [t ₁, σ _i,1). The amount of service received by flow n from server i during [t ₁, t ₂), W _{n, i} (t ₁, t ₂) is greater than or equal to:

$$\begin{array}{@{}rcl@{}} W_{n,i}(t_{1},t_{2}) & \ge & \rho_{n,i}(t_{1},\sigma_{i,1}) +\rho_{n,i}(t_{1}) - \rho_{n,i}(\sigma_{i,1})\\ & \ge & \rho_{n,i}(t_{1},\sigma_{i,1}) - L , \end{array} $$

(71)

where the second inequality follows from the fact that ρ _{n, i} (t ₁) ≥ 0 and ρ _{n, i} (σ _i,1) < L. Tokens allocated from server i during [t ₁, σ _i,0] are completely released before σ _i,1. Hence,

$$\begin{array}{@{}rcl@{}} \rho_{n,i}(t_1,\sigma_{i,1}) &\ge& \Theta_{n,i}(t_1,\sigma_{i,0}^+)\\ &\ge& \Theta_{n,i}(t_1,\sigma_{0}^+), \end{array} $$

(72)

where σ ₀ := min i{σ _i,0|δ _{n, i} = 1}. The lower bound in Eq. 72 along with Eq. 71 result in:

$$\begin{array}{@{}rcl@{}} W_n(t_1,t_2) & = & \sum\limits_{i:\delta_{n,i}=1}W_{n,i}(t_1,t_2)\\ &\ge& \sum\limits_{i:\delta_{n,i}=1}[\Theta_{n,i}(t_1,\sigma_{0}^+)-L]\\ & = & \Theta_{n}(t_1,\sigma_{0}^+) - L\sum\limits_i\delta_{n,i} \end{array} $$

(73)

Assume that token allocations are updated l times during [t ₁, σ ₀]. Specifically, let τ ₁, τ ₂, ... , τ _l ∈ [t ₁, σ ₀] denote the updating instants. Hence:

$$ \Theta_{n}(t_1,\sigma_{0}^+) = \sum\limits_{h=1}^l\Theta_{n}(\tau_h,\tau_h^+) $$

(74)

where $\Theta _{n}(\tau _{h},\tau _{h}^{+})$ is the amount of tokens allocated to flow n in the h ^thupdate, 1 ≤ h ≤ l. Since some flowsmay become inactive during (τ _h−1, τ _h ), it follows that:

$$\begin{array}{@{}rcl@{}} \Theta_{n}(\tau_h,\tau_h^+) & = & \sum\limits_i\alpha_{n,i}^{(h)}\hat{s}_i(\tau_h)\\ &\ge& \sum\limits_i\alpha_{n,i}^{(h)}{s}_i(\tau_h), \end{array} $$

(75)

where $\{\alpha _{n,i}^{(h)}\}$are the allocation parametersfor the h ^thupdate. We define theassigned service rate to flow n, r _n (τ _h ) as in Eq. 66. According to Lemma 4 (c.f. Eq. 70):

$$\begin{array}{@{}rcl@{}} \sum\limits_i\alpha_{n,i}^{(h)}s_i(\tau_h) & = & r_n{(\tau_h)}[\tau_h-\tau_{h-1}]\\ &\ge& {\int}_{\tau_{h-1}+\bar{T}}^{\tau_h+\bar{T}}\hat{r}_n(z)dz, \end{array} $$

(76)

where τ ₀is defined as thelast instant of time before t ₁, τ ₀ < t ₁, at whichtoken allocations are updated. The inequality in Eq. 76 along with Eqs. 74 and 75 result in:

$$\begin{array}{@{}rcl@{}} \Theta_{n}(t_1,t_2) &\ge& \sum\limits_{h=1}^l{\int}_{\tau_{h-1}+\bar{T}}^{\tau_h+\bar{T}}\hat{r}_n(z)dz\\ & = & {\int}_{\tau_{0}+\bar{T}}^{\tau_l+\bar{T}}\hat{r}_n(z)dz \end{array} $$

(77)

$$\begin{array}{@{}rcl@{}} &\ge& {\int}_{t_1+\bar{T}}^{\tau_l+\bar{T}}\hat{r}_n(z)dz \end{array} $$

(78)

where the last inequality follows from the fact that τ ₀ < t ₁.We prove that $\tau _{l}\ge t_{2}-2\bar {T}$. Toobserve this, let i ^∗ := arg min i{σ _i,0|δ _{n, i} = 1}. Tokensallocated at τ _l , are releasedat the next cycle on server i ^∗ starting at $\sigma _{i^{*},0}$. This cycle continues to $\sigma _{i^{*},1}$. Thenext cycle starting at $\sigma _{i^{*},1}$continues past t ₂. The fact that server i ^∗is continuously active over $(t_{2}-2\bar {T},t_{2}]$, implies that $\sigma _{i^{*},1}> t_{2}-\bar {T}$and$\tau _{l}\ge \sigma _{i^{*},1}-\bar {T}$(c.f. Eq. 30). It follows that $\tau _{l}\ge t_{2}-2\bar {T}$. The resulting lower bound from Eq. 78 along with Eq. 73 imply that:

$$ W_n(t_1,t_2) \ge {\int}_{t_1+\bar{T}}^{t_2-\bar{T}}\hat{r}_n(z)dz - L\sum\limits_i\delta_{n,i}. $$

(79)

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