Queuing Systems: Lecture 6 Amedeo odoni October 22. 2001
Queuing Systems: Lecture 6 Amedeo R. Odoni October 22, 2001
Lecture outline A fundamental result for queuing networks State transition diagrams for Markovian ueuing systems and networks: example Analysis of systems with dynamic demand and service rates Qualitative behavior of dynamic systems Reference: Sections 4.10 and 4.11
Lecture Outline • A fundamental result for queuing networks • State transition diagrams for Markovian queuing systems and networks: example • Analysis of systems with dynamic demand and service rates • Qualitative behavior of dynamic systems • Reference: Sections 4.10 and 4.11
A result which is important in analyses of queuing networks Let the arrival process at a M/Mm queuing system with infinite queue capacity have parameter n. Then, under steady state conditions(<mu) the departure process from the queuing system is also Poisson with parameterλ Implication: greatly facilitates analysis of open acyclic networks consisting of M/M/m queues with infinite queue capacities The bad news: result holds only under exact set of conditions described above
A result which is important in analyses of queuing networks Let the arrival process at a M/M/m queuing system with infinite queue capacity have parameter l. Then, under steady state conditions (l<mm) the departure process from the queuing system is also Poisson with parameter l. Implication: greatly facilitates analysis of open acyclic networks consisting of M/M/m queues with infinite queue capacities. The bad news: result holds only under exact set of conditions described above
Open acyclic network of M/M systems #2:=2 A=43 1 negative omental server P=1/3 =4#1:=3(pel A=43 server) M/M/2 Q=2/3 #3:=6 1 negative 入=4 =8/3 exponential server
Open acyclic network of M/M/. systems #1: m=3 (per server) M/M/2 l=4 #2: m=2 1 negative exponential server #3: m=6 1 negative exponential server P=1/3 Q=2/3 l=4/3 l=4/3 l=8/3 l=4
State transition diagrams for queuing systems and networks When external arrivals are poisson and service times are negative exponential many complex queuing systems and open acyclic queuing networks can be analyzed, even under dynamic conditions, through a udicious choice of state representation This involves writing and solving(often numerically) the steady-state balance equations or the Chapman-Kolmogorov first-order differential equations The"hypercube model"(Chapter 5 is a good example)
State transition diagrams for queuing systems and networks • When external arrivals are Poisson and service times are negative exponential, many complex queuing systems and open acyclic queuing networks can be analyzed, even under dynamic conditions, through a judicious choice of state representation • This involves writing and solving (often numerically) the steady-state balance equations or the Chapman-Kolmogorov first-order differential equations • The “hypercube model” (Chapter 5 is a good example)