What version of Sendmail are you using? You want Sendmail 8.12.4 or later. You should consider running Sendmail in deliverymode=queue with frequent queue runs if you want the best possible performance. (Take a look sendmail/TUNING in the Sendmail distribution.)
You should also configure Sendmail as described in the Cyrus documentation, using LMTP delivery to Cyrus. You don't want "deliver" processes---it's a completely unnecessary fork. There's an Addison Wesley book on "sendmail performance tuning" soon to come out that might help you with this. I'm going to be receiving a review copy and I'll report on it. Larry