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3 trauma bays. Trauma center moves to diversion status once all servers are busy (incoming patients are directed to other locations). Figure 7.1.: Process flow diagram for trauma center. Given P m (r) we can compute: Time per day that system has to deny access

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3 trauma bays

Trauma center moves to diversion status once all servers are busy (incoming patients are directed to other locations)

Figure 7.1.: Process flow diagram for trauma center


Analyzing loss systems finding p m r

  • Given Pm(r) we can compute:

  • Time per day that system has to deny access

  • Flow units lost = 1/a * Pm (r)

Analyzing Loss Systems: Finding Pm(r)

m

  • Define r = p / a

  • Example: r= 2 hours/ 3 hours r=0.67

  • Recall m=3

  • Use Erlang Loss Table

  • Find that P3 (0.67)=0.0255

r = p / a


Erlang loss table
Erlang Loss Table

Probability{all m servers busy}=


0.6

Probabilitythat all serversare utilized

0.5

0.4

m=1

m=2

0.3

0.2

m=3

m=5

m=10

0.1

m=20

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Implied utilization

Figure 7.2.: Implied utilization vs probability of having all servers utilized


Probabilitythat system isfull, Pm

Percentageof demand rate

100

0.5

80

0.4

Increasing levels

of utilization

60

0.3

Increasing levels

of utilization

40

0.2

20

0.1

0

0

1

2

3

4

5

6

7

8

9

10

11

1

2

3

4

5

6

7

8

9

10

11

Size of the buffer space

Size of the buffer space

Figure 7.3: Impact of buffer size on the probability Pm for various levels of implied utilization as well as on the throughput of the process in the case of one single server


Fraction of

customer lost

Average wait time [seconds]

Figure 7.4.: Impact of waiting time on customer loss


Outflow of resource 1 =

Inflow of resource 2

Inflow

Outflow

Upstream

Downstream

Figure 7.5.: A serial queuing system with three resources


Resource is blocked

Activity completed

Inflow

Outflow

Outflow

Inflow

Activity not

yet completed

Resource is starved

Empty space for a flow unit

Space for a flow unit with a flow unitin the space

Figure 7.6.: The concepts of blocking and starving


6.5 min/unit

6.5 min/unit

6.5 min/unit

7 min/unit

7 min/unit

7 min/unit

6 min/unit

6 min/unit

6 min/unit

Sequential system, no buffers

Cycle time=11.5 minutes

Sequential system, unlimited buffers

Cycle time=7 minutes; inventory “explodes”

3 resources, 19.5 min/ unit each

(1)

(1)

Horizontally pooled system

Cycle time=19.5/3 minutes=6.5 minutes

Sequential system, one buffer space each

Cycle time=10 minutes

Figure 7.7.: Flow rate compared at four configurations of a queuing system


Waitingproblem

Loss

problem

Pure waitingproblem, all customersare perfectly patient.

All customers enter the process,some leave due totheir impatience

Customers do notenter the process oncebuffer has reached a certain limit

Customers are lostonce all servers arebusy

Same if customers are patient

Same if buffer size=0

Same if buffer size is extremely large

Figure 7.8.: Different types of variability problems


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