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1Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Polling: Lower Waiting Time, Longer Processing Time (Perhaps)Waiting Lines
2Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Now Let’s Look at the Rest of the System; The Little’s Law Applies Everywhere
Flow time T = Ti + Tp
Inventory I = Ii + Ip
R
I = R T
R = I/T = Ii/Ti = Ip/Tp
Ii = R Ti Ip = R Tp
We know that U= R/RpWe have already learned Rp = c/Tp, R= Ip/TpWe can show U= R/Rp = (Ip/Tp)/(c/Tp) = Ip/cBut it is intuitively clear that U = Ip/c
3Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Variability in arrival time and service time leads to
Idleness of resources
Waiting time of flow units
Characteristics of Waiting Lines
We are interested in two measures
Average waiting time of flow units in the waiting line and in the system (Waiting line + Processor).
Average number of flow units waiting in the waiting line (to be then processed).
4Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Operational Performance Measures
Flow time T = Ti + Tp
Inventory I = Ii + Ip
Ti: waiting time in the inflow buffer = ?
Ii: number of customers waiting in the inflow buffer =?
Given our understanding of the Little’s Law, it is then enough to know either Ii or Ti.
We can compute Ii using an approximation formula.
5Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Utilization – Variability - Delay Curve
VariabilityIncreases
Average time in system
Utilization U100%
Tp
T
6Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Our two measures of effectiveness (average number of flow units waiting and their average waiting time) are driven by Utilization: The higher the utilization the longer
the waiting line/time. Variability: The higher the variability, the longer
the waiting line/time. High utilization U= R/Rp or low safety capacity Rs
=Rp – R, due to High inflow rate R Low processing rate Rp = c/Tp, which may be
due to small-scale c and/or slow speed 1/Tp
Utilization and Variability
7Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Variability in the interarrival time and processing time is measured using standard deviation (or Variance). Higher standard deviation (or Variance) means greater variability.
Standard deviation is not enough to understand the extend of variability. Does a standard deviation of 20 represents more variability or a standard deviation of 150
Drivers of Process Performance
for an average
for an average of 1000?
of 80
Coefficient of Variation: the ratio of the standard deviation of interarrival time (or processing time) to the mean(average).
Ca = coefficient of variation for interarrival time
Cp = coefficient of variation for processing time
8Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
U= R /Rp, where Rp = c/Tp
Ca and Cp are the Coefficients of Variation
Standard Deviation/Mean of the inter-arrival or processing times (assumed independent)
The Queue Length Approximation Formula
U1U 1)2(c
Utilization effect
U-part
Variability effect
V-part
2
22pa CC
iI
9Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Utilization effect; the queue length increases rapidly as U approaches 1.
Factors affecting Queue Length
U1U 1)2(c
iI
Variability effect; the queue length increases as the variability in interarrival and processing times increases.
While the capacity is not fully utilized, if there is variability in arrival or in processing times, queues will build up and customers will have to wait.
2
22pa CC
10Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Coefficient of Variations for Alternative Distributions
Tp: average processing time Rp =c/Tp
Ta: average interarrival time Ra = 1/Ta
Sp: Standard deviation of the processing time
Sa: Standard deviation of the interarrival time
Interarrival Time or Processing Time distributionMean Interarrival Time or Processiong TimeStandard Deviation of interarrival or Processing TimeCoeffi cient of Varriation of Interarrival or Processing Time
General (G)
Tp (or Ta)
Sp (or Sa)
Sp/Tp (or Sa/Ta)
Poisson (M)
Tp (or Ta)
Tp (or Ta)
1
Exponential (M)
Tp (or Ta)
Tp (or Ta)
1
Constant (D)
Tp (or Ta)
0
0
11Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Ta=AVERAGE () Avg. interarrival time = 6 min.Ra = 1/6 arrivals /min. Sa=STDEV() Std. Deviation = 3.94 Ca = Sa/Ta = 3.94/6 = 0.66
Coefficient of Variation
Example. A sample of 10 observations on Interarrival times in minutes 10,10,2,10,1,3,7,9, 2, 6 min.
Example. A sample of 10 observations on Processing times in minutes 7,1,7, 2,8,7,4,8,5, 1 min.
Tp= 5 minutes; Rp = 1/5 processes/min.
Sp = 2.83
Cp = Sp/Tp = 2.83/5 = 0.57
12Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Utilization and Safety Capacity
On average 1.56 passengers waiting in line, even though U <1 and safety capacity Rs = RP - Ra= 1/5 - 1/6 = 1/30 passenger per min, or 60(1/30) = 2/hr.
U1U 1)2(c
2
22pa CC
iI 0.831
0.831)2(1
2
57.066.0 22
Example. Given the data of the previous examples.
Ta = 6 min Ra=1/6 per min (or 10 per hr).
Tp = 5 min Rp =1/5 per min (or 12 per hr).
Ra< Rp R=Ra . U= R/ RP = (1/6)/(1/5) = 0.83
Ca = 0.66, Cp =0.57
56.1
13Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Waiting time in the line? RTi = Ii
Ti=Ii/R = 1.56/(1/6) = 9.4 min.
Waiting time in the system?
T = Ti+Tp
Since Tp = 5 T = Ti+ Tp = 14.4 min.
Total number of passengers in the system?
I = RT = (1/6) (14.4) = 2.4
Alternatively, 1.56 are in the buffer. How many are with the processor?
I = 1.56 + 0.83 = 2. 39
Example: Other Performance Measures
14Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Compute R, Rp and U: Ta= 6 min, Tp = 5 min, c=2R = Ra= 1/6 per minuteProcessing rate for one processor 1/5 for two processorsRp = 2/5U = R/Rp = (1/6)/(2/5) = 5/12 = 0.417
Now suppose we have two servers
On average Ii = 0.076 passengers waiting in line.Safety capacity is Rs = RP - R = 2/5 - 1/6 = 7/30
passenger per min or 60(7/30) = 14 passengers per hr or 0.233 per min.
U1U 1)2(c
2
22pa CC
iI 0.4171
0.4171)2(2
2
57.066.0 22
15Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Ti=Ii/R = (0.076)(6) = 0.46 min.
Total time in the system:
T = Ti+Tp
Since Tp = 5 T = Ti + Tp = 0.46+5 = 5.46 min
Total number of passengers in the process:
I = 0.076 in the buffer and 0.417 in the process.
I = 0.076 + 2(0.417) = 0.91
Other Performance Measures for Two Servers
c U Rs Ii Ti T I1 0.83 0.03 1.56 9.38 14.38 2.4
2 0.417 0.23 0.077 0.46 5.46 0.91
16Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Terminology: The characteristics of a waiting line is captured by five parameters; arrival pattern, service pattern, number of server, queue capacity, and queue discipline. a/b/c/d/e
Terminology and Classification of Waiting Lines
M/M/1; Poisson arrival rate, Exponential service times, one server, no capacity limit.
M/G/12/23; Poisson arrival rate, General service times, 12 servers, queue capacity is 23.
17Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
Exact Ii for M/M/c Waiting Line
18Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
The M/M/c Model EXACT Formulas
Steady-State, Infinite Capacity Queues
Basic Inputs: Number of Servers, c = 2
Arrival Rate, R i = 10
Service Rate of each server, 1/T p = 12
The Waiting Line: Average Number Waiting in Queue (I i ) = 0.17507
Average Waiting Time (T i ) = 0.01751
Q: Probability of more than 20 customers waiting = 0%
T: Probability of more than 0.5 time-units waiting = 0.02%
Service: Average Utilization of Servers = 41.67%
Average Number of Customers Receiving Service (I p) = 0.8333
The Total System (waiting line plus customers being served):Average Number in the System (I ) = 1.008
Average Time in System (T ) = 0.1008
Model is OK
19Ardavan Asef-Vaziri Oct. 2011Operations Management: Waiting Lines 1
The M/M/c/b Model EXACT Formulas
Steady-State, Finite Capacity Queues
Basic Inputs: Number of Servers, c = 6Queue Capacity, K = 6
Arrival Rate, R i = 5
Service Rate Capacity of each server, R p = 0.92308
Arrivals: Average Rate Joining System (R ) = 4.69048
Average Rate Leaving Without Service (R iP b) = 0.30952
Customers who Balk: Probability that System is Full (P b) = 6.19%
The Waiting Line: Average Number Waiting in Queue (I i ) = 1.560
Average Waiting Time (T i ) = 0.33261
Q: Probability of more than 0 customers waiting = 48.7%
Service: Average Utilization of Servers = 84.69%
Average Number of Customers Being Served (I p) = 5.08135
The Total System (waiting line plus customers being served):Average Number in the System (I ) = 6.641
Average Time in System (T ) = 1.41595
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