# Calculating Waiting Times and Waiting Time Distribution

by Hubert Becker

## 1. General

Spare parts are considered as an important factor for the effectivity of a support system. The provision and maintenance of an appropriate stock of spares is decisive for the availability of a technical system.

One effectivity measurement for an inventory system is the waiting time for spare parts. In general, the mean or average waiting time is an applicable measurement of effectivity. In order to provide a warranted performance and to determine the risk, the maximum waiting time and the waiting time distribution have to be known.

## 2. Calculation of Mean Waiting Times

The average number of demands during a given time T is equal to the expression lT and can be calculated by the following formula:
 (1)

 NSys : n : QPS : r Item : MTBA : T : No. of Systems Utilisation Rate (e.g. flying hours per system and calendar hour) Quantity per System Replacement Rate Mean Time Between Arisings Repair Turn Around Time (including Transportation Times) [hours]

The average, or mean, waiting time (MWT) is directly related to the Expected Number of Backorders (Expected Backorders - EBO). Backorders are demands that can not be satisfied immediately. These demands will be satisfied, when a spare part is coming out of the repair cycle. An inventory system with backorder case is shown in  fig. 1.

Within a considered representative period of time T four demands of each one spare part occur: demand D1 through D4. At the time S1 to S4 one repaired spare comes out of the repair cycle. Hence, the stock level is 1 during the times t1, t7 and t9, during time t8 the stock level is 2, and during the remaining times t2 to t6 no spare parts are available.

Fig. 1: Example for an inventory system with backorder case

The spare part’s demand at the point in time D2 is delayed and can be satisfied only after arrival of an item at time S1. Likewise the demand at time D3 can be satisfied only at time S2. In the time period T, as shown in fig. 1, two demands are backordered. The measure of "Expected Backorders " is an over the time weighed average value within the time period T and can be determined as follows:
 EBO = [1 × (t3+t4) + 1 × (t4+t5)]/T = (t3+ 2 × t4+t5)/T (2)

The average number of backorders also can also be described with the help of Markov Chains with the following states:

S, S-1, S-2, S-k ,...,S-(S-1), S-S, S-(S+1), -2,..., -¥

In the states S to S-(S-1) all demands can be fulfilled. Starting from status S-(S+1), first 1 demand, then 2, etc., can not be fulfilled directly. The backorders demands can be calculated therefore as total of all probabilities w(k) for the states k ³ S+1, multiplied by the term (k-s); the general equation reads:
 (3)

Assuming a Poisson distribution for wk(k|lT) the equation for the calculation of the Expected Backorders reads:
 (4)

For practical calculations the above equation is not suitable because of the Summation to infinite. As derived in [1] the average number of backorders can be determined by machine more easily with finite Summation using the following formula:
 (5)

The Mean Waiting Time (MWT) can be calculated from the Expected Number of Backorders (EBO) by division by the demand rate l:
 (6)

The Mean Waiting Time considers all demands, even those, for which the waiting time because of available stock is zero. In the case of demands, which can not be satisfied immediately, longer waiting will occur than the Mean Waiting Time.

In fig. 2 is shown the relation of MWT to the Repair Turn Around Time T as a function of the expected value of  lT and number of spares S. The Mean Waiting Time can be calculated thus by multiplication with the Repair Turn Around Time T.

Fig. 2 :  Diagram for the determination of the Mean Waiting Time (MWT) with Poisson demand [2]

If no spare parts are available, the relation of the Mean Waiting Time MWT to the Repair Turn Around Time T is always one by definition, i.e. the Mean Waiting Time is equal to the Repair Turn Around Time. Further it is evident that with a longer Repair Turn Around Time more spare parts are needed, in order to achieve the same absolute value for the waiting time.

## 3. Distribution of Waiting Times

In chapter 2 the formula for the calculation of the Mean Waiting Time was derived. It was also shown, that the maximum waiting time is equal to the Repair Turn Around Time, if no spare parts are stocked. With a certain probability, the waiting time for spare parts will be zero. This probability is equal to the Fill Rate (FR), for which the Demand Satisfaction Rate is a synonym.

The Fill Rate formula given Poisson distributed demand reads:
 (7)

with
 T : l : S : Repair Turn Around Time (including Transportation Times) [hours] average demand rate per hours No. of spares

The diagram in fig. 3 may be used to determine the Fill Rate corresponding to formula (7) as a function of the expected value of lT and the number of spares S.

Fig. 3: Diagram for the determination of the Fill Rate (FR) with Poisson demand [2]

Considering a deterministic (constant) Repair Turn Around Time T and Poisson distributed demands the cumulative waiting time distribution reads [3]:
 (8)

In fig. 4 the distribution of waiting time is shown by the ratio WT/T for varied

• demand rates l and

• different stock levels

(Click to enlarge)

Fig. 4:  Distribution of waiting time

The cumulative probability for the ratio WT/T=0 is per definition equal to the Fill Rate. For a ratio WT/T=1 the maximum waiting time will occur. Between those two points, the waiting time distribution is a function of the demand rate l and the stock level S.

Considering low demand rates (0 <l £ 3) and higher stock levels the waiting time distribution is approximately linear. For higher demands (l > 3) and higher stock levels the shape of waiting time distribution is concave, for lower stock levels it is convex.

In order to achieve acceptable support performance, a Fill Rate (WT=0) of at least 0.8, better 0.85 is required. As shown in fig. 4, in the latter case a concave waiting time distribution was obtained and might be approximated by a linear interpolation; the solution will be at the safe side in this case.

## 4. References

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