Matakuliah
Tahun
: K0414 / Riset Operasi Bisnis dan Industri
: 2008 / 2009
Model Antrian Ganda
Pertemuan 21
Learning Outcomes
• Mahasiswa akan dapat menghitung penyelesaian model
antrian tunggal dan ganda dalam berbagai contoh aplikasi.
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Outline Materi:
• Model Antrian Ganda M/M/C
• Jaringan Antrian
• Contoh Penerapan
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M/M/S Model
• Type: Multiple servers; single-phase.
• Input source: Infinite; no balks, no reneging.
• Queue: Unlimited; multiple lines; FIFO (FCFS).
• Arrival distribution: Poisson.
• Service distribution: Negative exponential.
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M/M/S Equations
Probability of zero people
or units in the system:
P0 
1
M 1 1  λ  n  1  λ  M Mμ
       
 n 0 n!  μ   M!  μ  Mμ  λ


M
Average number of people
or units in the system:
  

 
Ls 
P
M  !M    
Average time a unit spends
in the system:
  

 
Ws 
P 
M  !M   

M
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Note: M = number of servers in these equations
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M/M/S Equations
Average number of people

or units waiting for service: Lq  Ls 

Average time a person or
unit spends in the queue:
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
Wq  Ws 

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M/M/2 Model Equations
Average time in system:
Average time in queue:
Ws =
4
42 - 2
2

Wq =
 (2 + )(2 -)
Average # of customers in queue: Lq =  W
q
Average # of customers in system:
Ls =  W s
Probability the system is empty: P = 2 -
0 2 + 
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M/M/2 Example
Average arrival rate is 10 per hour. Average service
time is 5 minutes for each of 2 servers.
 = 10/hr,  = 12/hr, and S=2
Q1: What is the average wait in the system?
Ws =
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412
= 0.1008 hours = 6.05 minutes
2
2
4(12) -(10)
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M/M/2 Example
 = 10/hr,  = 12/hr, and S=2
Q2: What is the average wait in line?
(10)2
Wq =
= 0.0175 hrs = 1.05 minutes
12 (212 + 10)(212 - 10)
Also note:
1
Ws = Wq + 
1
so Wq = Ws  = 0.1008 - 0.0833 =0.0175 hrs
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M/M/2 Example
 = 10/hr,  = 12/hr, and S=2
Q3: What is the average number of customers in line
and in the system?
L q =  W q = 10/hr  0.0175 hr = 0.175 customers
L s =  W s = 10/hr  0.1008 hr = 1.008 customers
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M/M/2 Example
 = 10/hr and  = 12/hr
Q4: What is the fraction of time the system is empty
(server is idle)?
212 - 10
P0 =
= 41.2% of the time
212 + 10
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M/M/1, M/M/2 and M/M/3
1 server
2 servers
3 servers
25 min.
1.05 min.
0.1333 min. (8 sec.)
0.417 hr
0.0175 hr
0.00222 hr
WS
30 min.
6.05 min.
5.1333 min.
Lq
4.167 cust.
0.175 cust.
0.0222 cust.
LS
5 cust.
1.01 cust.
0.855 cust.
P0
16.7%
41.2%
43.2%
Wq
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Service Cost per Day
 = 10/hr and  = 12/hr
Suppose servers are paid $7/hr and work 8 hours/day and
the marginal cost to serve each customer is $0.50.
M/M/1 Service cost per day
= $7/hr x 8 hr/day + $0.5/cust x 10 cust/hr x 8 hr/day

= $96/day
M/M/2 Service cost per day
= 2 x $7/hr x 8 hr/day + $0.5/cust x 10 cust/hr x 8 hr/day
= $152/day
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Customer Waiting Cost per Day
 = 10/hr and  = 12/hr
Suppose customer waiting cost is $10/hr.
M/M/1 Waiting cost per day
= $10/hr x 0.417 hr/cust x 10 cust/hr x 8 hr/day = $333.33/day
M/M/1 total cost = 96 + 333.33 = $429.33/day
M/M/2 Waiting cost per day
= $10/hr x 0.0175 hr/cust x 10 cust/hr x 8 hr/day =$14/day
M/M/2 total cost = 152 + 14 = $166/day
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Unknown Waiting Cost
Suppose customer waiting cost is not known = C.
M/M/1 Waiting cost per day
= Cx 0.417 hr/cust x 10 cust/hr x 8 hr/day = 33.33C $/day
M/M/1 total cost = 96 + 33.33C
M/M/2 Waiting cost per day
= Cx 0.0175 hr/cust x 10 cust/hr x 8 hr/day =1.4C $/day
M/M/2 total cost = 152 + 1.4C
M/M/2 is preferred when 152 + 1.4C < 96 + 33.33C or
C > $1.754/hr
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M/M/2 and M/M/3
Q: How large must customer waiting cost be for M/M/3 to be
preferred over M/M/2?
M/M/2 total cost = 152 + 1.4C
M/M/3 Waiting cost per day
= Cx 0.00222 hr/cust x 10 cust/hr x 8 hr/day = 0.1776C $/day
M/M/3 total cost = 208 + 0.1776C
M/M/3 is preferred over M/M/2 when
208 + 0.1776C < 152 + 1.4C
C > $45.81/hr
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Remember:
 &  Are Rates
  = Mean number of arrivals per
time period.
– Example: 3 units/hour.
If average service time is 15
minutes, then μ is 4
customers/hour
  = Mean number of arrivals
served per time period.
– Example: 4 units/hour.
• 1/ = 15 minutes/unit.
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Other Queuing Models
• M/D/S
– Constant service time; Every service time is the same.
– Random (Poisson) arrivals.
• Limited population.
– Probability of arrival depends on number in service.
• Limited queue length.
– Limited space for waiting.
• Many others...
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