1. Model Antrian M/G/1

2. The M/G/1 Model Assumptions
Interarrival times have an exponential distribution with a mean of 1/l.
Service times can have any probability distribution. You only need the mean (1/m) and standard deviation (s).
The queueing system has one server.
The probability of zero customers in the system is
P0 = 1 – r
The expected number of customers in the queue is
Lq = [l2s2 + r2] / [2(1 – r)]
The expected number of customers in the system is
L = Lq + r
The expected waiting time in the queue is
Wq = Lq / l
The expected waiting time in the system is
W = Wq + 1/m
© The McGraw-Hill Companies, Inc., 2003

3. Steady-State Parameters of M/G/1 Queue
r = l / m
L = r + {l2 (m-2 + s2)} / {2 (1 - r)} = r + {r2 (1 + s2 m2)} / {2 (1 - r)}
W = m-1 + {l (m-2 + s2)} / {2 (1 - r)}
Wq = {l (m-2 + s2)} / {2 (1 - r)}
Lq = {l2 (m-2 + s2)} / {2 (1 - r)} = {r2 (1 + s2 m2)} / {2 (1 - r)}
P0 = 1 - r
19 February 2019
R. Rumani M / Telkom Institute of Technology 30 16

4. The Values of s and Lq for the M/G/1 Model with Various Service-Time Distributions
Mean s
Model Lq
Exponential
1/m
M/M/1
r2 / (1 – r)
Degenerate (constant)
M/D/1
(1/2) [r2 / (1 – r)]
Erlang, with shape parameter k
(1/k) (1/m)
M/Ek/1
(k+1)/(2k) [r2 / (1 – r)]
© The McGraw-Hill Companies, Inc., 2003
Table The values of s and Lq for the M/G/1 model with various service-time distributions.

5. Example 1 Baker and Able
There are two workers competing for a job. Able claims an average service time which is faster than Baker’s, but Baker claims to be more consistent, if not as fast. The arrivals occur according to a Poisson process at a rate of l= 2 per hour. (1/30 per minute). Able’s statistics are an average service time of 24 minutes with a standard deviation of 20 minutes. Baker’s service statistics are an average service time of 25 minutes, but a standard deviation of only 2 minutes. If the average length of the queue is the criterion for hiring, which worker should be hired?
19 February 2019
R. Rumani M / Telkom Institute of Technology 30 13

6. Example 1 Baker and Able For Able, l = 1/30 (per min), m-1 = 24 (min)
r = l / m = 24/30 = 4/5
s2 = 202 = 400(min2)
Lq = {l2 (m-2 + s2)} / {2 (1 - r)}
= {(1/30)2 ( )} / {2 (1-4/5)}
= (customers)
For Baker
l = 1/30 (per min), m-1 = 25 (min)
r = l / m = 25/30 = 5/6
s2 = 22 = 4(min2)
Lq = {(1/30)2 ( )} / {2 (1-5/6)}
= (customers)
19 February 2019
R. Rumani M / Telkom Institute of Technology 30 16

7. Example 1 Baker and Able
Although working faster on the average, Able’s greater service variability results in an average queue length about 30% greater than Baker’s. On the other hand, the proportion of arrivals who would find Able idle and thus experience no delay is P0 = 1 - r = 1 / 5 = 20%, while the proportion who would find Baker idle and thus experience no delay is P0 = 1 - r = 1 / 6 = 16.7%. On the basis of average queue length, Lq , Baker wins.
19 February 2019
R. Rumani M / Telkom Institute of Technology 30 13

8. Example 2 VP of Engineering Approach
© The McGraw-Hill Companies, Inc., 2003
l = 3
The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq ≤ 1/4 day).
The Vice President for Engineering has suggested providing tech reps with new state-of-the-art equipment that would reduce the time required for the longer repairs.
After gathering more information, they estimate the new equipment would have the following effect on the service-time distribution:
Decrease the mean from 1/4 day to 1/5 day.
Decrease the standard deviation from 1/4 day to 1/10 day.

9. Example 2 VP of Engineering Approach
© The McGraw-Hill Companies, Inc., 2003
Figure This Excel template for the M/G/1 model shows the results from applying this model to the approach suggested by the Dupit’s vice president for engineering to use state-of-the-art equipment.

10. Standar Deviasi Jika Standar Deviasi tidak diketahui, maka
𝜎= 1 𝑁−1 𝑖=1 𝑁 𝑥 𝑖 − 𝑥 2
N = jumlah data
𝑥 = mean

11. Example 3 sampai 10:30 Minimarket dengan 1 Kasir
Hitunglah :
Standar Deviasi waktu pelayanan.
Rata-rata banyaknya pelanggan dalam Minimarket (Ls)
Rata-rata pelanggan yang mengantri (Lq)
Berapa lama pelanggan mengantri (Wq)
Berapa lama pelanggan dilayani oleh Kasir (Wb = Ws-Wq)