LED Blinker Duty Cycle
This multivibrator circuit of yours has only two states.  In order to get what you ask for, you need more states.  You likely will have to build it using several monostable multvibrators, and have each one trigger the next, in a loop.

Edit: Since you posted a photo, I see it is a purchased item.  Nonetheless, you can modify it to be monostable, and you can buy several boards to achieve your goal.

The standard answer is to use a microcontroller to do this instead, but I generally believe in answering the original question and letting you decide.

Details:
With this two transistor circuit, there are three ways to wire it:


Bistable: Whichever light is on, stays on. You have to input something (like push a switch) to make the other light come on instead.  Then it stays that way. It can be flipped the other way with another switch. This behavior is like the switch that turns on the lights in the room.
Monostable: Using a switch, you can "kick" it to the other state, where it sits for a given time, then flips back to where it was, and stays that way.  This is comparable to a 3-minute egg timer.
Astable: It stays in one state for a time, then flips to the other.  The other state is timed, and flips back to the first state.  It never stops.  This is commonly called a "flasher".  


In this circuit, either the left transistor is on, or the right one is.  When the left side turns off, the right side turns on.  The corresponding LED's indicate this.  It's really flashing one LED, and the second one can be added, but it will always be in the opposite state from the first.  These things happen together, so it isn't possible to have both off at the same time.  That would require third state--and more parts.

Let's say you could build a three-state circuit.  You'd have these states:


Left LED on, right LED off.
Right LED on, left LED off.
Both LED's off.


Question: When both lights are off, which one is going to come on next?  Your desired circuit has two places where one of the LED's go off without the other coming on. So, really, you need four states to produce this effect:


Left LED on, right LED off.
Both LED's off, with right one next.
Right LED on, left LED off.
Both LED's off, with left one next.


You can use a 555 in place of the multivibrator.  And it can be wired to be astable, bistable or monostable.  But,  you still would need four of them. Both kinds of circuits can provide a pulse to trigger the next circuit, and both have an input where the trigger can be fed in.

I suggest looking at the diagram in the Wikipedia article on multivibrators (or maybe you got a schematic with the board?), and find all the like parts on your board.  If you didn't get a schematic, you should try to draw one.