Workload and when to actually ask for more people

Maybe you could have coffee talk with your boss regarding the matter. Present to him a comprehensive report conclusive of the impact of the things left unfinished vis-a-vis the timeline. Then maybe your boss can do the recommendation. 

How much work-load is it?  How much can be expected?

A published example, from (Cypress Semiconductor CEO) TJ Rodgers in his book _No_Excuses_Management_ (Doubleday, 1992, ISBN 0-385-42604-6, p164).  Here's how one design manager justified proposed head-count.  He took data on how many engineers it took to design each chip, the size of each chip, and how many designers on each chip -- "In other words, he created a design-productivity index [measured in transistors per designer per day].  Then he extrapolated: He used that index to determine how many designers he needed based on the complexity of his chip and its target delivery date.  That manager was making a bet: 'I plan to manage my project as productively as any new-product project in this company's history.  If you insist that I hire fewer people, then you are betting I will set a new record for productivity [...] but if you lose that bet, the design schedule will slip and the launch will be delayed.'  I approved his request on the spot, no questions asked."

An example from my own experience is, during one period of overload, I listed out each thing that my manager had asked me to handle.  I listed it all, with time estimates based on previous durations, and prioritized it.  Came out to 14 hours a day, when I could only do [rather] fewer.  Each morning, I made a fresh copy of the list, summarizing hours outstanding, marking off each thing as it was completed, along with citing when it was promised and when it was delivered.  After a few weeks, my boss started complaining about what was dropping, so I went over the list and reprioritized.  After another two weeks, my boss started getting complaints about dropped things -- and he was already keenly aware I was working flat-out (from my timesheets) on only what was most valuable (from hours backlogged per my prioritized progress sheets).  After another few weeks, the CEO called him on the carpet for all the dropped stuff costing us revenue and profit.  He had copies of my sheets showing that I needed to be cloned.  The CEO delivered me two new team-mates to train three weeks later.

It's not all wins, though.  At another company I worked for, several of us went into similar overload.  It got worse as more and more of us dropped away, no-one being added despite plenty of data about the lost profits.  Turns out a "vulture capitalist" had taken over and was simply draining the company for people and equipment (and tax losses) to use elsewhere before flipping it in bankruptcy.  After the final firings, I took two months before I felt ready to find the next job.

I found that mostly it was my boss that suggested adding more people. The key to it was in the weekly professional update (or one-on-one if you are lucky) you are regularly updating on your team's achievements, explain how you are delegating, what you are prioritizing/letting go, and how the team are managing the workload. If you are patient you may find after a few updates your boss may ask what difference it would make with an extra person or whether you think you need an extra person. That is then your good opportunity to expand.

I am in a position now. I was asked to justify adding another team member by my boss. With some inspiration from MT I have begun to draft a document demonstrating what another person can do for the company rather than what they would do for me.