Maximum Throughput

Appendix This appendix contains some hints on the calculation of the performance values mentioned in section 3.2. The probability that submitted queries (updates) can be executed at the local database is denoted as l q (l u respectively) and results in l q loc loc r qr = + − ⋅ ⋅ () 1 and l u loc = because loc expresses the preference of accessing original local data and the second term reflects the local read availability introduced by replication. Note that replication does not increase the write availability because of the primary copy approach. The overall rate of queries to be executed at a local database (λ q total) not only consists of queries submitted by users but also of additional queries received from other sites: λ λ λ q total q q q q n n = ⋅ + − ⋅ − ⋅ ⋅ − l l () () 1 1 1 1 = ⋅ + − ⋅ = l l q q q q q λ λ λ () 1 Considering the identical behavior of sites λ λ q total q = is not much of a surprise: every site receives just as many queries as it forwards to other sites due to a lack of appropriate local data. For updates we similarly derive λ λ λ λ u total u u u u u n n n r k ur = ⋅ + − ⋅ − ⋅ ⋅ − + − ⋅ ⋅ ⋅ ⋅ l l () () () 1 1 1 1 1 = + − ⋅ ⋅ ⋅ ⋅ (()) 1 1 n r k ur u λ In addition to the rate of locally submitted updates (λ u) the amount of propagated updates has to be included: An arbitrary update must be propagated with probability r⋅k⋅ur at each of the n-1 remaining sites. The combined flow of arriving queries and updates at each local database system is still Poisson with rate λ λ λ λ λ total q total u total q u n r k ur = + = + + − ⋅ ⋅ ⋅ ⋅ (()) 1 1 Since the service time of the combined stream is hyperexponentially distributed and each node acts like a M/H 2 /1/FCFS system the average waiting time W at a local database can be derived using the Pollaczek-Khinchin formula [24]: W t t …