The battery life can be calculated by determining the average current for the circuit.
Use the following general equation to calculate average current.
Iavg = (Ton*Ion + Tsleep*Isleep ) / (Ton +Tsleep)
Ton (arduino is active ) = 250 ms =0.25s and Ion = 16mA
Tsleep = 5min =300s and Isleep = 200 uA (approx)
It is very difficult measure the current during the sleep period .Some time I got zero reading.
Iavg = 0.205 mA
Operating Voltage =5 V
Pavg=VxIavg =5x.205=1.026 mW
Li Ion battery capacity =3000 mAh
Battery voltage =3.7V
Power =3.7x3000=11100 mWh
Battery life = 11100/1.026 =10,818.7 hours = 15 months approximately
From the above calculation it is clear that theoretically by using a fully charged 3000 mAh Li Ion battery we can run the arduino for 15 months.In practical due to battery self discharge this figure may be different.
As the system is equipped with solar charging system ,we can run for few years without any interruption.