\section{Theoretical Analysis}\label{sec:CR}\label{sec:5}

In this section, we analyze the competitive ratio of the OWA algorithm and show that it is tight and approaches the maximum possible competitive ratio as the budget $K$ increases. Before that, we first show that the optimization problem $\mathtt{P}_1$ can be solved by standard convex optimization solvers.
\begin{theorem}\label{trm:cov}
    (Convex Equivalence) There exists a convex optimization problem $\mathtt{P}'$, such that its optimal solution is also the optimal solution to $\mathtt{P}_1$.
\end{theorem}
\begin{proof}
    \emph{(Sketch of Proof)} We first transform the objective function into an equivalent form. We then show that there exists an optimal solution that satisfies the quality of a key constraint, so we can further transform the problem into a convex optimization problem. Consequently, the optimal solution of this convex problem is also optimal for $\mathtt{P}_1$. The complete proof is provided in Appendix~\ref{sec:trm:cov}.
\end{proof}


\subsection{Competitive Ratio of the OWA Algorithm}

To analyze the competitive ratio of the OWA algorithm, we will lower-bound the probability of processing each task $t$ by each worker $l$ by the assignment guarantee $\gamma_l^*$. To show that OWA avoids the situation where the tasks can no longer be accepted because the remaining amount of resource falls below 1, leading to a $0$ probability of processing subsequent tasks, we proactively identify a ``safe range'' for $\{\gamma_l^*\}$. This ensures that the resource remains sufficiently distant from the stop line. We start with a definition of resource sufficiency in Definition \ref{def:1}.

\begin{definition}\label{def:1}
     We say that the resource sufficiency condition is satisfied if the following statement is true: For each task $t$, $K-\Theta_{t}\geq1$ if $\Theta_{t}\leq\theta_{t}$. 
\end{definition}

The meaning of Definition \ref{def:1} is that, if the current resource consumption $\Theta_{t}\leq \theta_{t}$, we guarantee that the remaining amount of the resource is greater than or equal to $1$, allowing processing of the future tasks.
We next show that the computed $\gamma_l^*$ from solving $\mathtt{P}_1$ in Eqs.~(\ref{eq:p1start})--(\ref{eq:p1end}) satisfies the resource sufficiency condition in Definition~\ref{def:1}.

\begin{theorem}\label{trm:1}
    (Sufficient Resource) With $\gamma_l^*$ from solving $\mathtt{P}_1$ in Eqs.~(\ref{eq:p1start})--(\ref{eq:p1end}), the OWA algorithm satisfies the resource sufficiency condition.

\end{theorem}
\begin{proof}
\emph{(Sketch of Proof)} To prove Theorem~\ref{trm:1}, we require several lemmas. First, we define $H_t$ as the anti-derivative of $h_t$ and show $H_{t+1}(\theta_t)\leq H_t(\theta_t)$ for every $t\in[T]$. Second, we show that $\theta_t\leq\theta_{t+1}$ for every $t$. Third, we define $\overline{\Theta}_t=\min_w\{\int_{-\infty}^wh_t(v)dv=1\}$ and show that $\overline{\Theta}_{t+1}\leq\theta_t+1$ for every $t$. Fourth, we show that $\theta_t\leq\theta_{t+1}<\theta_t+1$ for every $t$.  Finally, we show that constraint~(\ref{eq:9}) implies that $K\geq\theta_t+1$, which in turn leads to the resource sufficiency condition. The complete proof is given in Appendix~\ref{sec:trm1}.
\end{proof}

As a result of Theorem~\ref{trm:1}, OWA will accept a task and process it by some worker $l$ as long as the current resource consumption $\Theta_{t}$ is less than or equal to the target $\theta_{t}$. Using such an approach, we can bound from below the probability that each task $t$ will be processed by each worker $l$.

\begin{theorem}\label{trm:2}
    (Worker Assignment Probability Lower Bound) For all $l$, the probability of processing each task $t$ by worker $l$ is no less than $\gamma_l^*$. 
\end{theorem}
\begin{proof}
\emph{(Sketch of Proof)} To prove Theorem~\ref{trm:2}, we first show a relation between $h_t(w)$ and the distribution of $\Theta_t$ over all sample paths. Using this relation, the design of the OWA algorithm, the update rule of $h_t(w)$, and Theorem~\ref{trm:1}, we arrive at the theorem's conclusion. The complete proof is given in Appendix~\ref{sec:trm2}.
\end{proof}

Theorem~\ref{trm:2} leads to the competitive ratio of OWA:
\begin{theorem}\label{trm:3}
    (Competitive Ratio) The competitive ratio achieved by the OWA algorithm is at least
    % \begin{align}
        $\alpha=\mathtt{P}_1(\{\gamma_l^*\})$.
    % \end{align}
\end{theorem}
\begin{proof}
 \emph{(Sketch of Proof)} We use the results of Theorem~\ref{trm:2} to analyze the competitive ratio of the OWA algorithm. We find that when the probability of processing each task $t$ by worker $l$ is lower bounded by $\gamma_l^*$, the competitive ratio of the OWA algorithm is at least $\mathtt{P}_1(\{\gamma_l^*\})$. The complete proof is given in Appendix~\ref{sec:trm:3}.
\end{proof}

Since in general we only have a numerical solution to the optimization problem $\mathtt{P}_1$ and its optimal value $\mathtt{P}_1(\{\gamma_l^*\})$, we give a closed-form lower bound on the competitive ratio $\mathtt{P}_1(\{\gamma_l^*\})$ in the following corollary:

\begin{corollary}\label{crly:1}
    (Closed-Form Lower Bound on the Competitive Ratio) The competitive ratio of OWA is lower bounded by 
    % \begin{align}
        % $\alpha'={(1-K^{-\frac{1}{2}})}/{L}$.
        $\alpha'=\max\{{1}/{L},c\}\cdot(1-K^{-\frac{1}{2}})$, where $c=\min_{l\in[L]}\underline{u}_l/\max_{l\in[L]}\overline{u}_l$ is a constant derived from the problem instance.
\end{corollary}
\begin{proof}
\emph{(Sketch of Proof)} To prove Corollary~\ref{crly:1}, we find a feasible solution to the optimization problem $\mathtt{P}_1$ and show that the competitive ratio achieved by this solution in GMPMW is no less than $\alpha'$. Then, since $\alpha=\mathtt{P}_1(\{\gamma_l^*\})$ is the optimal value obtained on problem $\mathtt{P}_1$ by the optimal solution $\gamma_l^*$, it is lower bounded by $\alpha'$. The complete proof is given in Appendix~\ref{sec:crly1}.
\end{proof}
Note that the numerical value of $\alpha$ consistently exceeds its lower bound $\alpha'$. We provide numerical results in  Appendix~\ref{sec:crly1:num} to illustrate the actual behavior of the competitive ratio $\alpha$.
When there is only one worker (i.e., $L=1$), the competitive ratio $\alpha$ is lower bounded by $\alpha'=1-K^{-\frac{1}{2}}$, matching the previous best result on the original single-worker GMP~\citep{alaei2013online}. This result confirms that our algorithm design is consistent with the previous best algorithm design for the single-worker GMP, while also addressing the multiple workers in GMPMW.


\subsection{Tightness of the Competitive Ratio}
Now that we have obtained the competitive ratio of the OWA algorithm, we investigate how tight this performance bound is. 
First, we give a formal definition of the tightness of the competitive ratio of an online algorithm in Definition~\ref{def:2}.

\begin{definition}\label{def:2}
    (Tightness) The competitive ratio $\alpha$ for an online algorithm $\text{ALG}$ is tight if and only if there exists a task sequence $I$ where
    % \begin{align}
        $\mathbb{E}_{i\sim I}[\text{ALG}(i)]= \alpha \max_{i\in \Omega^I}[\text{OPT}(i)]$.
    % \end{align}
\end{definition}

Next, we prove that the competitive ratio $\alpha=\mathtt{P}_1(\{\gamma_l^*\})$ achieved by the OWA algorithm on GMPMW is tight.

\begin{theorem}\label{trm:4}
(Tightness of $\alpha$) For every combination of $K$ and $L$, there exists a tasks sequence $I$, where the performance ratio achieved by the OWA algorithm is exactly $\mathbb{E}_{i\sim I}[\text{ALG}(i)]/\max_{i\in \Omega^I}[\text{OPT}(i)]=\alpha=\mathtt{P}_1(\{\gamma_l^*\})$.
\end{theorem}
\begin{proof}
We focus on the task sequence where the reward lower and upper bounds for each $l$ are $\underline{u}_l=0$ and $\overline{u}_l=u$. In this case, the competitive ratio of OWA is $\alpha=\mathtt{P}_1(\{\gamma_l^*\})=\min_l \gamma_l^*$. 
To induce the worst-case performance, the adversary will arrange the task sequence in the following way: The reward for processing each task $t$ by the worker $l'=\arg\min_l\gamma_l^*$ is set to $u_{t,l'}=u$, and $u_{t,l}=0$ for processing each task by every other worker $l$. The offline optimal result is thus $\max \text{OPT}(i)=Tu$, which is obtained when the offline algorithm processes each task $t$ by worker $l'$, reached when $\sum_{t\in[T]}r_{t,l'}<K$. In this case, the expected total reward obtained by the OWA algorithm is exactly $\gamma_{l'}^*Tu$, resulting in a performance ratio of $\gamma_{l'}^*=\min_l \gamma_l^*=\mathtt{P}_1(\{\gamma_l^*\})$.
\end{proof}

\subsection{Optimality of the OWA Algorithm}
We investigate whether any online algorithm can achieve a higher competitive ratio on GMPMW than that of OWA. For the single-worker GMP,~\citet{alaei2013online} proved the existence of asymptotically optimal solutions only for the special case where the reward lower bound is $0$.
In what follows, we show that the competitive ratio $\alpha$ of OWA is also asymptotically optimal with respect to $K$ when the reward lower bound $\underline{u}_l$ for every worker $l$ is $0$ (i.e., $c=0$), such that OWA recovers the best existing solution for GMP~\citep{alaei2013online} when there is only one worker (i.e., $L=1$).
However, since $\alpha$ does not have a closed-form expression, we will first prove that its lower bound $\alpha'$ is asymptotically optimal, as stated in Theorem~\ref{trm:5}.

\begin{theorem}\label{trm:5}
    (Asymptotic Optimality of $\alpha'$) When the reward lower bound of each worker is $0$, $\alpha'$ is an asymptotically optimal competitive ratio for the OWA algorithm on GMPMW, meaning that $\alpha'$ approaches the maximum possible competitive ratio when budget $K$ approaches infinity.
\end{theorem}
\begin{proof} 
    (Sketch of Proof) To prove Theorem~\ref{trm:5}, we show that there exists an adversary for which no online algorithm can achieve a competitive ratio higher than $\alpha^*=\frac{1}{L}$.
    The complete proof is in Appendix~\ref{sec:trm5}.
\end{proof}
\begin{remark}
    Since Corollary~\ref{crly:1} establishes that $\alpha' \leq \alpha$, Theorem~\ref{trm:5} directly implies that, when $c = 0$, the competitive ratio $\alpha$ of OWA also approaches the maximum competitive ratio as the budget $K$ approaches infinity.
\end{remark}



Note that the asymptotic optimality of~\citep{alaei2013online} was also proved in the case where the reward lower bound is $0$, such that our result implies that OWA is a natural generalization of the existing best solution to incorporate the multiple workers in GMPMW. In Appendix~\ref{sec:A.D.1}, we further provide numerical results to study the impact of various system parameters on the competitive ratio of OWA.
