Model Antrian M/G/1

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

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 / rrm@ittelkom.ac.id Telkom Institute of Technology 30 16

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 14.6 The values of s and Lq for the M/G/1 model with various service-time distributions.

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 / rrm@ittelkom.ac.id Telkom Institute of Technology 30 13

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 (242 + 400)} / {2 (1-4/5)} = 2.711 (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 (252 + 4)} / {2 (1-5/6)} = 2.097 (customers) 19 February 2019 R. Rumani M / rrm@ittelkom.ac.id Telkom Institute of Technology 30 16

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 / rrm@ittelkom.ac.id Telkom Institute of Technology 30 13

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.

Example 2 VP of Engineering Approach © The McGraw-Hill Companies, Inc., 2003 Figure 14.7 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.

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

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)