A detailed review of inventory models involving positive service time is given. These include classical and retrial cases. Also contributions to production inventory with service time is indicated towards the end. In addition directions for future work are indicated.

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The paper deals with a continuous review bulk demand (positive-integer valued) ( s, S) inventory system where the interarrival times of demands are independent and identically distributed random variables. We assume that the successive quantities demallded lie between a and b (0< a⩽ b; s- b+l⩾0) with pk, k= a, a+l, ... , b-1, b as the probability that k items are demanded by an arriving unit. The maximum capacity of the system is S units and as soon as the inventory level falls to the set A= { s- b + 1, s- b + 2, ... , s -1, s}, order is placed for a quantity S- i if the ordering level is i, i ε A. Our model assumes that the quantity replenished forms a Markov chain defined over thestatespace E={ c, c + l, ... , S - s} with c⩾ b. Lead time is zero and no shortage is permitted. The distribution of the on band inventory at arbitrary time point and also the limiting distributions are obtained. A numerical illustration associated with the model is also provided.