《土木与环境工程》（英文版） Spatially distributed Queues

Spatially Distributed Queues M/G/1 2 Servers n servers approximations

Spatially Distributed Queues M/G/1 2 Servers N servers Approximations

MG1 Directions Of travel 0,Y0 口 Ambulance (0,0

M/G/1 Ambulance (0,0) (0,Y0) (X0,0) Directions Of Travel XA YA

M/G1 Directions Of travel 0,Y Ambulance

M/G/1 Ambulance (0,0) (0,Y0) Directions Of Travel XA YA (X0,0)

MG1 0 Ambulance always returns home with each service; standard M/G/1 applies a But suppose we have an emergency repair vehicle that travels directly from one customer to the next?

M/G/1 aAmbulance always returns home with each service; standard M/G/1 applies aBut suppose we have an emergency repair vehicle that travels directly from one customer to the next?…

MG1 S,OS, expected value and variance, respectivel of the lst service time in a busy period 230S expected value and variance, respectively, of the 2nd all succeeding service times in a busy period p=l-P= fraction of time server is busy

M/G/1 S1 of the 1st service time in a busy period S2 of the 2nd λ ρ ,σ S1 2 = expected value and variance,respectively, ,σ S22 = expected value and variance,respectively, & all succeeding service times in a busy period S2 < 1 =1 − P0 = fraction of time server is busy

MG1 p=- 3+S+λ⑧s(3+3)-5(053+S p+ (S2-S) (1-S2)

M/G/1 ρ = λS1 1 − λ(S2 − S1 ) L = ρ + λ2 1 − λ(S2 − S1 )[σ S12 + S12 + λ{S1(σ S 2 2 + S22 ) − S 2 (σ S1 2 + S12 ) 2(1 − λ S2) ]

MG1 Little s law: buy one get three others for free! L=n See the book, eqs. (5.0)-(5.5)

M/G/1 Little' s Law :Buy one,get three others for free! L = λW L q = λWq See the book, Eqs. (5.0) - (5.5)

Two-Server“ Hypercube Queueing Model a Distinguishable servers a Different workloads(due to geography) a Can appear with or without queueing AWith--usually FCFS AWithout--usually means a backup contract service is in place

Two-Server “Hypercube” Queueing Model aDistinguishable servers aDifferent workloads (due to geography) aCan appear with or without queueing `With -- usually FCFS `Without -- usually means a backup contract service is in place

B-A B="Service region

A B-A B = “Service Region

Poisson arrivals from any sub-region a 入(B-A)入2 B-A (4)=A1 B="Service region =x1+A2

Poisson Arrivals from any sub-region A A B-A λ ( A)= λ1 λ (B-A)= λ 2 B = “Service Region” λ = λ1+ λ 2