1. Extension of the multi-class APQ with heterogeneous servers:
We have presented the expressions in closed form of the waiting time distributions and the conservation law in a multi-class heterogeneous multi-server APQ. We anticipate two major directions for future developments:
• Heterogeneous multi-server APQ optimization problems, such as, how to select the parameter set{bk;k = 1, . . . ,K}to minimize an appropriately defined cost or meet certain waiting time targets, as well as optimal control of heterogeneous APQs using different service dispatch policies.
• Extension to the situation where the service time distributions are non-exponential. Such a problem is non-trivial, although some results might be possible in the case of the service time distributions built upon the exponential. Other types of service dispatch policies can be considered in such systems, for instance, the idle-time- based dispatch policies including Longest Idle Server First (LISF), and Shortest Idle Server First (SISF).
2. Accumulating priority queues with other types of priority accumulation functions: We have presented conditions for a general nonlinear APQ to have a linear proxy, in which customers of all classes have the same waiting time distribution as in the nonlinear APQ. There are two major sets of priority accumulation functions left to be explored:
6.2. Future work 129
• An APQ with affine priority accumulation functions, in which customers of differ- ent classes start with different amounts of initial priority.
• An APQ with more general nonlinear priority accumulation functions, for which a linear proxy does not exist.
3. Multi-classM/Mi/cAPQs with class-dependent service rates:
We are currently studying a multi-classM/Mi/cAPQ where the service rate depends on the customer’s class type, rather than the server. However, this problem is complicat- ed by the possible combinations of customers’ types in service at the beginning of an accreditation interval. Hopefully, we may solve the problem in the near future.
4. Extension of the optimization problems of queues operating under waiting time limits: We have formulated and studied the WAE and IWAE optimization problems for the sys- tems operating under waiting time limits, where the investigations and derivations are primarily considered for the two-class multi-server APQs in this thesis. We have made a further exploration of the case with multiple classes of customers. Based on some fundamental derivations we have carried out, we foresee that explicit solutions would be obtained for the IWAE in the multi-class queues.
Additional materials in Chapter 3
A.1
Waiting times of accredited customers in an
M/M
i/cAPQ
Stanfordet al. [24] introduces the concept of the “the maximum priority process”{Mk(t);k =
1,2, . . . ,K}. Fork = 1,2, . . . ,K, Mk(t) provides an upper bound on the possible accumulated priority of class-k customers at time t. Within a single busy period, class-k (and higher) cus- tomers can eventually exceed the maximum priority of customers from classes with priorities lower thank. Such customers are said to be “served at accreditation levelk” if their priority up- on entry to service at timetlies in the interval (Mk+1(t),Mk(t)]. Moreover, the authors defined
the accreditation interval at levelk, which is a period of time that starts either at the beginning of a busy period or when a customer is served at some accreditation level l1 for l1 > k, and
finishes either at the end of a busy period or when another customer is served at some accredi- tation levell2forl2 > k. Whenever a customer is served at accreditation levelk, accreditation
intervals at all levell< kcommence.
Stanfordet al. [24] showed that an accreditation interval at levelk can be thought of as a delay cycle in the sense of Conwayet al. [4] that starts with the service time of the initiating customer and continues as long as there are customers at accreditation levell≤k. In our case, within the delay cycle, the service times are exponentially distributed with rate µa, and the instants at which customers of all classesi;i≤kbecome accredited at levelkare distributed as
AppendixA.1: Waiting times of accredited customers in an M/Mi/cAPQ 131 a Poisson process with rateΛk = P
k
i=1λi(bi−bk+1)/bi. Therefore, the LST ˜Γk(s) of the duration
of the accreditation interval at levelkis obtained by solving the functional equation ˜
Γk(s)= B˜(s+ Λk(1−Γ˜k(s))) (A.1) (see Conway et al. [4, page 150, equation (7)]), where ˜B(s) = µa/(µa+ s), which results in equation (3.28).
Letσk denote the stationary proportion of time that the server spends on customers served at all accreditation levelsl= 1,2, . . . ,k, so thatσk = Pkj=1ρj(bj−bk+1)/bj (see Stanfordet al. [24]). Then the LST of the waiting time of a class-kcustomer who is served at accreditation levelk; W˜acc((k) s), is shown in Stanfordet al.[24] to be given by
˜ Wacc(k)(s)= 1−ρ 1−σk +W˜ (k+1) + bbk+1 k s;µ,c ! k X j=1 ρj(bk+1/bj) 1−σk + K X j=k+1 ρj (1−σk)W˜ (j) + bj bk s;µ,c ! ˜ Wacc(k,0)(s). (A.2) The term ˜Wacc(k,0)(s) in equation (A.2) is given by
˜ Wacc(k,0)(s)= [µΓk−1 − Pk i=1λi(bk−bk+1)/bi][ ˜φk(sbk+1/bk)−Γ˜k−1(s)] (1−bk+1/bk)[s−(P k i=1λibk/bi)(1−Γ˜k−1(s))] , (A.3)
where 1/µΓk is the mean of a random variable with LST ˜Γk(s); that is, 1/µΓk = − dΓ˜k(s)
ds |s=0. The LST, ˜φk(s), is the solution to the functional equation ˜φk(s)= Γ˜k−1[s+(Pki=1λi(bk−bk+1)/bi)(1−
˜
φk(s))]. The derivation of these expressions can be found in Stanfordet al.[24].
Remark Assumingbk+1/bk = 1/M; k = 1,2, . . . ,K − 1. Under the classical priority queu- ing discipline, lim
M→∞bk+1/bk = 0, then we have limM→∞Λk =
Pk
i=1λi, limM→∞σk = Pkj=1ρj, and
lim
M→∞µΓk−1 = µa− Λk−1. By equations (3.31), (A.2) & (A.3), the LST of the delayed waiting
time distribution for each class under the classical priority queue is
lim M→∞ ˜ W+(k)(s;µ,c)= lim M→∞ ˜ Wacc(k)(s)= lim M→∞ ˜ Wacc(k,0)(s) = (µΓk−1 −λk) 1−Γ˜k−1(s) s−λk(1−Γ˜k−1(s)) = (µa−Λk) 1−Γ˜k−1(s) s−λk(1−Γ˜k−1(s)) . (A.4)
We observe that it can be shown readily that(A.4), after accounting for customers who do not wait, is consistent in the single server case with the unconditional waiting time LST found in Conwayet al.[4] page 164, equation (29).