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week_7_9-16_v2_hw.xls
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9_16
The wheat harvesting season in the American Midwest is short, and most farmers deliver their truck-loads of wheat to a giant central storage bin within a two-week span. Because of this, wheat-filled trucks waiting to unload and return to the fields have been known to back up for a block at the receiving bin. The central bin is owned cooperatively, and it is to every farmer's benefit to make the unloading/storage process as efficient as possible. The cost of grain deterioration caused by unloading delays, the cost of truck rental, and idle driver time are significant concerns to the cooperative members. Although farmers have difficulty quantifying crop damage, it is easy to assign a waiting and unloading cost for truck and driver of $ 18 per hour. The storage bin is open and operated 16 hours per day, 7 days per week, during the harvest season and is capable of unloading 35 trucks per hour according to an exponential distribution. Full trucks arrive all day long (during the hours the bin is open) at a rate of about 30 per hour, following a Poisson pattern. To help the cooperative get a handle on the problem of lost time while trucks are waiting in line or unloading at the bin?
(a) average number of trucks in the unloading system.
(b) average time per truck in the system
(c) utilization rate for the bin area.
(d) probability that there are more than three trucks in the system at any given time.
(e) total daily cost to the farmers of having their trucks tied up in the unloading process.
(f). The cooperative, as mentioned, uses the storage bin only two weeks per year. Farmers estimate that enlarging the bin would cut unloading costs by 50% next year. Would it be worth to enlarge storage area?
Arrival rate (λ) 30 Per day
Service rate (μ) 35 Per day
(a) average number of trucks in the unloading system.
Solution: computation of the following
Utilization rate(U)=λ/μ
server utilization (U) 85.71%
(b) average time per truck in the system
Solution: computation of the following
The average down time is the time that the machine waits to be serviced plus the time taken to repair the machine.
The average down time is given by W
W=1/1(μ-λ)
W 0.2 Day
assuming 8 hrs/day 1.6 Hours
(c) utilization rate for the bin area.
Solution: computation of the following
Lq=λ^2/μ(μ-λ)
Lq 5.142857143 Machines
(d) probability that there are more than three trucks in the system at any given time.
(e) total daily cost to the farmers of having their trucks tied up in the unloading process.
(f). The cooperative, as mentioned, uses the storage bin only two weeks per year. Farmers estimate that enlarging the bin would cut unloading costs by 50% next year. Would it be worth to enlarge storage area?
Solution: computation of the following
Pn>k=(λ/μ)^(k+1)
Pn>1 0.735
Pn>2 0.630
Pn>3 0.540
Pn>4 0.463

Solution_Excel modules

wheat harvesting
Queuing Model M/M/s (Exponential Service Times)
Input Data Operating Characteristics
Arrival rate (l) 30 Average server utilization (r) 0.8571
Service rate (m) 35 Average number of customers in the queue (Lq) 5.1429
Number of servers (s) 1 Average number of customers in the system (L) 6.0000
Average waiting time in the queue (Wq) 0.1714
Average time in the system (W) 0.2000
Probability (% of time) system is empty (P0) 0.1429
0
Probabilities
Number of Units Probability Cumulative Probability
0 0.1429 0.1429
1 0.1224 0.2653
2 0.1050 0.3703
3 0.0900 0.4602
4 0.0771 0.5373
5 0.0661 0.6034
6 0.0567 0.6601
7 0.0486 0.7086
8 0.0416 0.7503
9 0.0357 0.7859
10 0.0306 0.8165
11 0.0262 0.8427
12 0.0225 0.8652
13 0.0193 0.8845
14 0.0165 0.9010
15 0.0141 0.9151
16 0.0121 0.9272
17 0.0104 0.9376
18 0.0089 0.9465
19 0.0076 0.9542
20 0.0065 0.9607
Computations
n or s (lam/mu)^n/n! Cumsum(n-1) term2 P0(s) Rho(s) Lq(s) L(s) Wq(s) W(S)
0 1
1 0.8571428571 1 6 0.1428571429 0.8571428571 5.142857143 6 0.1714285714 0.2
2 0.3673469388 1.857142857 0.6428571429 0.4 0.4285714286 0.1928571429 1.05 0.0064285714 0.035
3 0.1049562682 2.224489796 0.1469387755 0.421686747 0.2857142857 0.0247848537 0.8819277108 0.0008261618 0.0293975904
4 0.0224906289 2.329446064 0.0286244368 0.4240755311 0.2142857143 0.0033106154 0.8604534726 0.0001103538 0.0286817824
5 0.0038555364 2.351936693 0.0046532336 0.4243419649 0.1714285714 0.0004085301 0.8575513873 0.0000136177 0.0285850462
6 0.0005507909 2.355792229 0.0006425894 0.4243698964 0.1428571429 0.0000454493 0.8571883064 0.000001515 0.0285729436
7 0.0000674438 2.35634302 0.0000768545 0.4243725877 0.1224489796 0.0000045509 0.8571474081 0.0000001517 0.0285715803
8 0.0000072261 2.356410464 0.0000080933 0.424372825 0.1071428571 0.0000004121 0.8571432693 0.0000000137 0.0285714423
9 0.0000006882 2.35641769 0.0000007606 0.4243728441 0.0952380952 0.000000034 0.8571428911 0.0000000011 0.0285714297
10 0.000000059 2.356418378 0.0000000645 0.4243728456 0.0857142857 0.0000000026 0.8571428597 8.55628E-11 0.0285714287
11 0.0000000046 2.356418437 0.000000005 0.4243728457 0.0779220779 0.0000000002 0.8571428573 5.95911E-12 0.0285714286
12 0.0000000003 2.356418442 0.0000000004 0.4243728457 0.0714285714 1.15423E-11 0.8571428572 3.84742E-13 0.0285714286
13 2.16477E-11 2.356418442 2.31758E-11 0.4243728457 0.0659340659 6.94247E-13 0.8571428571 2.31416E-14 0.0285714286
14 1.32537E-12 2.356418442 1.41181E-12 0.4243728457 0.0612244898 3.90738E-14 0.8571428571 1.30246E-15 0.0285714286
15 7.57353E-14 2.356418442 8.03254E-14 0.4243728457 0.0571428571 2.06593E-15 0.8571428571 6.88644E-17 0.0285714286
16 4.05725E-15 2.356418442 4.28691E-15 0.4243728457 0.0535714286 1.02976E-16 0.8571428571 3.43254E-18 0.0285714286
17 2.04567E-16 2.356418442 2.15429E-16 0.4243728457 0.0504201681 4.85428E-18 0.8571428571 1.61809E-19 0.0285714286
18 9.7413E-18 2.356418442 1.02284E-17 0.4243728457 0.0476190476 2.17032E-19 0.8571428571 7.2344E-21 0.0285714286
19 4.39457E-19 2.356418442 4.60219E-19 0.4243728457 0.0451127819 9.22698E-21 0.8571428571 3.07566E-22 0.0285714286
20 1.88339E-20 2.356418442 1.96772E-20 0.4243728457 0.0428571429 3.73901E-22 0.8571428571 1.24634E-23 0.0285714286
21 7.68729E-22 2.356418442 8.01441E-22 0.4243728457 0.0408163265 1.44728E-23 0.8571428571 4.82425E-25 0.0285714286
22 2.99505E-23 2.356418442 3.11647E-23 0.4243728457 0.038961039 5.36167E-25 0.8571428571 1.78722E-26 0.0285714286
23 1.11617E-24 2.356418442 1.15937E-24 0.4243728457 0.0372670808 1.90454E-26 0.8571428571 6.34848E-28 0.0285714286
24 3.98631E-26 2.356418442 4.13395E-26 0.4243728457 0.0357142857 6.49755E-28 0.8571428571 2.16585E-29 0.0285714286
25 1.36674E-27 2.356418442 1.41526E-27 0.4243728457 0.0342857143 2.1323E-29 0.8571428571 7.10766E-31 0.0285714286
26 4.50572E-29 2.356418442 4.65933E-29 0.4243728457 0.032967033 6.74077E-31 0.8571428571 2.24692E-32 0.0285714286
27 1.43039E-30 2.356418442 1.47729E-30 0.4243728457 0.0317460318 2.05548E-32 0.8571428571 6.85159E-34 0.0285714286
28 4.37874E-32 2.356418442 4.51702E-32 0.4243728457 0.0306122449 6.05336E-34 0.8571428571 2.01779E-35 0.0285714286
29 1.29421E-33 2.356418442 1.33363E-33 0.4243728457 0.0295566503 1.72372E-35 0.8571428571 5.74573E-37 0.0285714286
30 3.69774E-35 2.356418442 3.8065E-35 0.4243728457 0.0285714286 4.7511E-37 0.8571428571 1.5837E-38 0.0285714286
1. Both l and m must be RATES, and use the same time unit. For example, given a service time such as 10 minutes per customer, convert it to a service rate such as 6 per hour. 2. The total service rate (rate x servers) must be greater than the arrival rate.