This　article　investigates　the　issues　of　dissipativity　analysis　and　synthesis　of　discrete-time　switched　positive　systems,　in　the　presence　of　asynchronous　switching.　A　switching　retard,　between　activation　of　positive　subsystems　and　activation　of　candidate　controllers,　is　taken　into　account.　The　dwell-time　dependent　linear　copositive　storage　functions　and　linear　supply　rates　are　adopted　to　capture　the　property　of　dissipativity,　where　the　state　of　each　subsystem　is　required　to　stay　in　positive　orthant.　Sufficient　conditions　are　also　attained　such　that　the　closed-loop　system　with　time-varying　asynchronous　state-feedback　controllers　maintains　positivity　and　meets　strict　(q,r)-α-dissipative　performance.　Furthermore,　all　derived　criteria　are　presented　in　the　form　of　linear　vector　inequalities.　Finally,　two　examples　including　the　compartmental　model　are　offered　to　illustrate　the　validity　of　derived　results.　©　2023　Elsevier　Ltd